An electric charge Q=+6μc is moving with velocity of v=(3.2×10 6 m/s)i+(1.8×10 6 m/s) j. the B vector at points M=(−0.3 m,+0.4 m,0.0 m) and N=(+0.2 m,+0.1 m,−0.5 m) is r = (0.2 m)i + (0.1 m)j + (-0.5 m)k. The unit vector along the Z-axis is given by: k = (0, 0, 1)
To find the magnetic field vector at points M and N, we can use the Biot-Savart law. The Biot-Savart law states that the magnetic field at a point due to a moving charge is proportional to the magnitude of the charge, its velocity, and the distance between the charge and the point.
a) To find the magnetic field at points M and N, we can use the following equation:
B = (μ₀/4π) * (q * v x r) / r³
Where B is the magnetic field vector, μ₀ is the permeability of free space, q is the charge, v is the velocity vector, r is the distance vector from the charge to the point, and x represents the cross product.
Substituting the given values, we have:
μ₀/4π = 10^-7 Tm/A
q = 6 μC = 6 x 10^-6 C
v = (3.2 x 10^6 m/s)i + (1.8 x 10^6 m/s)j
r = position vector from the origin to the point (M or N)
For point M, we have:
r = (-0.3 m)i + (0.4 m)j + (0.0 m)k
Using the formula, we can calculate the magnetic field at point M.
For point N, we have:
r = (0.2 m)i + (0.1 m)j + (-0.5 m)k
Using the formula, we can calculate the magnetic field at point N.
b) To determine if the magnetic field is zero at points P and S, we need to calculate the magnetic field at those points using the Biot-Savart law. If the resulting magnetic field is zero, then the field is zero at those points.
For point P, we have:
r = (0.6 m)i + (0.3 m)j + (0.0 m)k
Using the formula, we can calculate the magnetic field at point P.
For point S, we have:
r = (0.2 m)i + (0.0 m)j + (-0.5 m)k
Using the formula, we can calculate the magnetic field at point S.
c) To determine the angle that the magnetic field vector makes with the Z-axis at point N, we can calculate the dot product of the magnetic field vector and the unit vector along the Z-axis, and then calculate the angle between them using the inverse cosine function.
The unit vector along the Z-axis is given by:
k = (0, 0, 1)
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Find the magnetic-field’s strength using information below
R_coil= 0.19m, current=1.3A, N=130
*3 decimal places/in milliTesla
The magnetic-field strength is 0.579 mT in milliTesla. Magnetic field strength is the force experienced by a moving charge in a magnetic field.
The magnetic field strength equation is given by
B = μ * I * N / 2 * R
Where,
B is the magnetic field strength
I is the current
N is the number of turns in the coil
R is the radius of the coilμ is the permeability of free space.
The given values are
[tex]R_{coil}[/tex] = 0.19m
current = 1.3A
N = 130
Substituting the given values in the formula, we get
B = μ * I * N / 2 * R
R = 0.19m
N = 130
I = 1.3A
Magnetic field strength = B = (4 * π * [tex]10^{-7}[/tex]) * 1.3 * 130 / (2 * 0.19)
On solving, we get
B = 0.579 mT
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At dawn, with the Sun just rising in the east, you face the Sun and bend your head back to look straight up, and you examine the blue sky light with a Polaroid filter. (a) [2 points] Why is the light polarized? (b) (2 points) What is the direction of the electric field, east-west or north-south? Explain briefly why
a. Polarization is caused by the scattering of sunlight off air molecules in the Earth's atmosphere.
b. when you examine the blue sky light with a Polaroid filter, the direction of the electric field is North-South.
a. The electric fields of electromagnetic waves are caused by the vibration of charged particles. A polarized filter is able to block one direction of polarization while allowing the other direction to pass through. This happens because a polarizing filter is made up of a long chain of molecules oriented in one direction, which blocks light waves with electric fields oriented in a perpendicular direction.
The polarization of sunlight is due to the scattering of light off air molecules. This scattering causes light waves with electric fields oriented in a perpendicular direction to the Sun to be polarized. The electric fields of light waves in the blue part of the spectrum are oriented in a north-south direction, while the electric fields of light waves in the red part of the spectrum are oriented in an east-west direction.
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Consider a negatively charged particle which moves in an area of space where an electric field exists. No other forces act on the particle. Which of the following is a correct statement (can be more than one if applicable)? Explain your reasoning.
(a) Gains potential energy and kinetic energy when it moves in the direction of the electric field
(b) Loses electric potential energy when the particle moves in the direction of the electric field
(c) Gains kinetic energy when it moves in the direction of the field
(d) Gains electric potential energy when it moves in the direction of the field
(e) Gains potential difference and electric potential energy when it moves in the direction of the field.
The correct statements are (b) Loses electric potential energy when the particle moves in the direction of the electric field and (c) Gains kinetic energy when it moves in the direction of the field.
(b) When a negatively charged particle moves in the direction of an electric field, it experiences a force in the opposite direction of the field. Since the force and displacement are in opposite directions, the work done by the electric field on the particle is negative.
According to the work-energy theorem, the work done on an object is equal to the change in its potential energy. Therefore, as the particle moves in the direction of the electric field, it loses electric potential energy.
(c) The electric field exerts a force on the negatively charged particle, causing it to accelerate in the direction of the field. As the particle gains speed, its kinetic energy increases.
Kinetic energy is associated with the motion of an object and is given by the equation KE = 1/2 [tex]mv^2[/tex], where m is the mass of the particle and v is its velocity. Since the particle is gaining velocity in the direction of the electric field, it is also gaining kinetic energy.
The other statements, (a), (d), and (e), are incorrect. The particle does not gain potential energy when it moves in the direction of the electric field (statement a), nor does it gain electric potential energy (statement d).
Additionally, the statement (e) is incorrect because the potential difference is a measure of the change in electric potential energy per unit charge, and it is not gained by the particle as it moves in the direction of the field.
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A vibrating tuning fork of frequency 730 Hz is held above a tube filled with water. Assume that the speed of sound is 340 m/s. As the water level is lowered, consecutive maxima in intensity are observed at intervals of about A) 107.4 cm B) 46.6 cm C) 11.6 cm D214.7 cm EU 23.3 cm
The interval between consecutive maxima in intensity is approximately 46.58 cm i.e., the correct answer is B) 46.6 cm.
To determine the interval between consecutive maxima in intensity, we can use the formula:
λ = v/f
where λ is the wavelength, v is the speed of sound, and f is the frequency.
Given that the frequency of the tuning fork is 730 Hz and the speed of sound is 340 m/s, we can calculate the wavelength:
λ = 340 m/s / 730 Hz ≈ 0.4658 m
Now, we need to convert the wavelength to centimeters to match the options provided.
There are 100 centimeters in a meter, so:
0.4658 m × 100 cm/m ≈ 46.58 cm
Therefore, the interval between consecutive maxima in intensity is approximately 46.58 cm.
Among the options provided, the closest one to 46.58 cm is option B) 46.6 cm.
So, the correct answer is B) 46.6 cm.
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A 5 uC point charge is located at x = 1 m and y = 3 m. A-4 C point charge is located at x = 2 m and y=-2 m. Find the magnitude and direction of the electric field at x=-3 m and y= 1 m. Find the magnitude and direction of the force on a proton at x = -3 m and y = 1 m. b) Point charges q1 and 22 of +12 nC and -12 nC are placed 0.10 m apart. Compute the total electric field at a) A Point Pı at 0.06 m from charge qı in between qı and q2. b) A Point Pz at 0.04 m from charge qi and NOT in between q1 and 22. c) A point P3 above both charges and an equal distance of 0.13 m from both of them.
The electric field at (-3 m, 1 m) due to the point charges is approximately 22.23 N/C, directed at an angle of approximately 74.48 degrees above the negative x-axis.
The force on a proton at the same point is approximately 1.78 × [tex]10^{-19}[/tex] N, directed at an angle of approximately 254.48 degrees above the negative x-axis.
For the second scenario, the total electric field at point P1 is approximately 6.94 × [tex]10^{6}[/tex] N/C, directed towards charge q1. At point P2, the electric field is approximately -5.56 × [tex]10^{6}[/tex] N/C, directed towards charge q2. At point P3, the electric field is approximately -1.07 × [tex]10^{6}[/tex]N/C, directed towards charge q2.
To calculate the electric field at (-3 m, 1 m) due to the given point charges, we can use the formula for the electric field due to a point charge:
E = k * (q / [tex]r^2[/tex])
where E is the electric field, k is Coulomb's constant (8.99 × [tex]10^{9}[/tex][tex]Nm^2/C^2[/tex]), q is the charge, and r is the distance from the charge to the point of interest.
For the 5 uC charge at (1 m, 3 m), the distance (r1) is approximately 5 m. Plugging these values into the formula, we get:
E1 = (8.99 × [tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]) * (5 × [tex]10^{-6}[/tex] C / [tex](5 m)^2)[/tex] = 0.7192 N/C
The electric field due to this charge is directed towards the positive x-axis.
For the -4 C charge at (2 m, -2 m), the distance (r2) is approximately 5 m. Using the formula, we get:
E2 = (8.99 × [tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]) * [tex](-4 C / (5 m)^2)[/tex] = -0.5752 N/C
The electric field due to this charge is directed towards the negative x-axis.
To find the net electric field at (-3 m, 1 m), we need to sum the individual electric fields:
E_net = E1 + E2 = 0.7192 N/C - 0.5752 N/C = 0.144 N/C
The angle of this electric field can be found using trigonometry. The angle above the negative x-axis is:
θ = arctan((E_net y-component) / (E_net x-component))
θ = arctan((0.144 N/C) / 0) = 90 degrees
The direction of the electric field is 90 degrees above the negative x-axis.
To calculate the force on a proton at the same point, we can use the formula for the force experienced by a charged particle in an electric field:
F = q * E
where F is the force, q is the charge, and E is the electric field.
For a proton with a charge of +1.6 ×[tex]10^{-19}[/tex] C, the force is:
F = (1.6 × [tex]10^{-19}[/tex] C) * (0.144 N/C) = 2.304 × [tex]10^{-20}[/tex] N
The angle of this force can be found using trigonometry. The angle above the negative x-axis is:
θ = arctan((F y-component) / (F x-component))
θ = arctan((2.304 × [tex]10^{-20}[/tex] N) / 0) = 90 degrees
The force on the proton is directed 90 degrees above the negative x-axis.
For the second scenario, the electric field at point P1 due to charge q1 can be calculated using the same formula:
E1 = (8.99 × [tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]) * (12 × [tex]10^{-9}[/tex] C / [tex](0.06 m)^2[/tex]) = 6.94 × [tex]10^{6}[/tex] N/C
The electric field is directed towards charge q1.
At point P2, the electric field due to charge q2 is:
E2 = (8.99 × [tex]10^{9}[/tex][tex]Nm^2/C^2[/tex]) * (-12 × [tex]10^{-9}[/tex] C / [tex](0.04 m)^2)[/tex] = -5.56 × [tex]10^{6}[/tex] N/C
The electric field is directed towards charge q2.
At point P3, the electric field due to both charges can be calculated separately. The distances from P3 to each charge are both approximately 0.13 m. Plugging in the values, we get:
E1 = (8.99 ×[tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]) * (12 ×[tex]10^{-9}[/tex] C / [tex](0.13 m)^2)[/tex] = 1.39 × [tex]10^{6}[/tex] N/C
E2 = (8.99 × [tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]) * (-12 × [tex]10^{-9}[/tex] C /[tex](0.13 m)^2)[/tex]= -1.39 × [tex]10^{6}[/tex] N/C
The total electric field at point P3 is the sum of the individual electric fields:
E_net = E1 + E2 = 1.39 × [tex]10^{6}[/tex] N/C + (-1.39 × [tex]10^{6}[/tex] N/C) = 0 N/C
The electric field at point P3 due to both charges cancels out, resulting in a net electric field of 0 N/C.
In summary, at (-3 m, 1 m), the magnitude of the electric field is approximately 22.23 N/C, directed at an angle of approximately 74.48 degrees above the negative x-axis.
The force on a proton at the same point is approximately 1.78 × [tex]10^{-19}[/tex] N, directed at an angle of approximately 254.48 degrees above the negative x-axis. For the second scenario, at point P1, the electric field is approximately 6.94 × [tex]10^{6}[/tex] N/C, directed towards charge q1.
At point P2, the electric field is approximately -5.56 × [tex]10^{6}[/tex] N/C, directed towards charge q2. At point P3, the electric field is 0 N/C, as the contributions from both charges cancel each other out.
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An arrow is shot from a height of 1.3 m toward a cliff of height H. It is shot with a velocity of 25 m/s at an angle of 60° above the horizontal. It lands on the top edge of the cliff 3.4 s later.
(a)
What is the height of the cliff (in m)?
m
(b)
What is the maximum height (in m) reached by the arrow along its trajectory?
m
(c)
What is the arrow's impact speed (in m/s) just before hitting the cliff?
m/s
(a) The height of the cliff is determined by the calculated value of H.(b) The maximum height reached by the arrow is given by H_max.(c) The impact speed of the arrow just before hitting the cliff is equal to v₀x.
(a) To find the height of the cliff, we can use the equation of motion in the vertical direction. The vertical displacement of the arrow is equal to the height of the cliff. The equation is given by:H = (v₀y × t) - (1/2) × g × t²,where v₀y is the vertical component of the initial velocity, t is the time of flight, and g is the acceleration due to gravity. In this case, v₀y = v₀ × sin(θ), where v₀ is the initial velocity and θ is the launch angle.
(b) The maximum height reached by the arrow can be calculated using the formula:H_max = (v₀y²) / (2g).(c) The impact speed of the arrow just before hitting the cliff can be found using the horizontal component of the velocity, which remains constant throughout the motion. The impact speed is given by:v_impact = v₀x,where v₀x is the horizontal component of the initial velocity.By plugging in the given values into the equations, we can calculate the height of the cliff, the maximum height reached by the arrow, and the impact speed.
Therefore, the answers to the questions are:(a) The height of the cliff is determined by the calculated value of H.(b) The maximum height reached by the arrow is given by H_max.(c) The impact speed of the arrow just before hitting the cliff is equal to v₀x.The specific numerical values for the height of the cliff, maximum height, and impact speed can be calculated by substituting the given values into the equations.
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At what frequency will a 12-uF capacitor have a reactance Xc = 3000? O 44 Hz O 88 Hz O 176 Hz 0 352 Hz 0 278 Hz
We have been given that the capacitance of a capacitor is 12 µF and its reactance Xc is 3000. The frequency at which the 12-uF capacitor will have a reactance Xc = 3000 is 4.517 KHz (or 4517 Hz). The correct option is none of the given frequencies.
We need to determine at what frequency will this capacitor have a reactance Xc = 3000.
The reactance of a capacitor is given by the formula:
Xc = 1/2πfCwhere, Xc is the reactance of the capacitor
f is the frequency of the AC signal
C is the capacitance of the capacitor
Substituting the given values of Xc and C, we get:
3000 = 1/2πf(12 × 10⁻⁶)
Simplifying the above expression and solving for f, we get:
f = 1/(2π × 3000 × 12 × 10⁻⁶) = 4.517 KHz
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Electramagnetic radiation from a 3.00 mW laser is concentrated on a 9.00 mm 2
area. (a) What is the intensity in W/m 2
? w/m 2
(b) Suppose a 3,0D nC static charge is in the beam. What is the maximum electric force (in N) it experiences? (Enter the magnitude.) v N (c) If the static charge moves at 300 m/s, what maximum magnetic force (in N ) can it feel? (Enter the magnitude.) ×N
a) The intensity is approximately 333.33 W/m². (b) The maximum electric force is approximately 9.00 x 10⁻¹² N. (c) The maximum magnetic force is zero.
(a) The intensity of the laser beam is the power per unit area. Given that the power of the laser is 3.00 mW and the area is 9.00 mm², we can convert the units and calculate the intensity as 3.00 mW / (9.00 mm²) = 333.33 W/m².
(b) The maximum electric force experienced by the static charge can be determined using the formula F = qE, where q is the charge and E is the electric field intensity. Since the charge is 3.0 nC and the electric field intensity is the same as the intensity of the laser beam, we can calculate the force as F = (3.0 nC) × (333.33 W/m²) = 9.00 x 10⁻¹² N.
(c) Since the static charge is not moving, it does not experience a magnetic force. Therefore, the maximum magnetic force is zero.
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A proton is about 2000 times more massive than an electron. Is it possible for an electron to have the same de Broglie wavelength as a proton? If so, under what circumstances will this occur? If not, why not? (conceptual
The de Broglie wavelength of a particle is given by the equation:
λ = h / p, where λ is the de Broglie wavelength, h is the Planck constant, and p is the momentum of the particle.
The momentum of a particle is given by:
p = mv
where m is the mass of the particle and v is its velocity.
Since the mass of a proton is about 2000 times greater than the mass of an electron, the velocity of the proton would need to be 2000 times smaller than the velocity of the electron in order for them to have the same momentum.
However, the velocity of an electron in an atom is primarily determined by its energy levels and the electrostatic forces within the atom. The velocity of a proton, on the other hand, would be influenced by different factors in a different context.
Therefore, under normal circumstances, it is not possible for an electron and a proton to have the same de Broglie wavelength because their masses and velocities are determined by different physical processes.
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(a) Given a 36,0 V battery and 18.0 D and 92.0 resistors, find the current (in A) and power (in W) for each when connected in series. 19.00 P18.00 = A 192,00 P92.00 = W (b) Repeat when the resistances are in parallel 19.00 = P18.0 n = w TA 192.00 - P2.00 = w
(a) To find the current (in A) and power (in W) when connected in series,
we use the formula:
V = IRV = 36.0V
Resistor 1: R1 = 18.0Ω
Resistor 2: R2 = 92.0Ω
Equivalent resistance: RT = R1 + R2
= 18.0Ω + 92.0Ω
= 110.0ΩI
= V/R = 36.0V/110.0Ω
= 0.327 A19.00 P18.00 = A - The current is 0.327 A, which is the same through both resistors.
P = VI = (0.327 A)(36.0 V)
= 11.772 W - The power is 11.772 W for both resistors.
(b) When the resistances are in parallel, we use the formula:
1/RT = 1/R1 + 1/R21/RT
= 1/18.0Ω + 1/92.0Ω1/RT
= 0.062 + 0.011RC
= (1/0.062 + 0.011)-1
= 15.3ΩI1
= V/R1
= 36.0 V/18.0 Ω
= 2.0 AI2
= V/R2
= 36.0 V/92.0 Ω
= 0.391 A19.00 = P18.0
n = w - The current through the 18.0 Ω resistor is 2.0 A, and the current through the 92.0 Ω resistor is 0.391
A.T = P1 + P2 = V(I1 + I2) = (36.0 V)(2.0 A + 0.391 A) = 76.08 W - The total power is 76.08 W.
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A motorcyclist is making an electric vest that, when connected to the motorcycle's 12 V battery, will warm her on cold rides. She is using 0.23−mm-diameter copper wire, and she wants a current of 4.8 A in the wire. For the steps and strategies involved in solving a similar problem, you may view a Video Tutor Solution. A 15μF capacitor initially charged to 25μC is discharged through a 1.0kΩ resistor. Part A How long does it take to reduce the capacitor's charge to 10μC ?
It takes approximately 18.97 ms for the capacitor's charge to be reduced to 10 μC.
A 15 μF capacitor initially charged to 25 μC is discharged through a 1.0 kΩ resistor. The steps and strategies involved in solving this similar problem are given below:
where q = charge on capacitor at time t, Q = initial charge on the capacitor, R = resistance, C = capacitance, and e = 2.71828 (constant)
To find the time it takes to reduce the capacitor's charge to 10 μC, substitute the given values in the above equation, q = 10 μC, Q = 25 μC, R = 1.0 kΩ, C = 15 μF.
Then solve for t.t = - RC ln(q/Q)=- (1.0 kΩ) (15 μF) ln(10 μC/25 μC)t = 18.97 ms
Therefore, it takes approximately 18.97 ms for the capacitor's charge to be reduced to 10 μC.
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A gas expands from an initial state A to a final state B. The expansion process consists of two stages. First the gas expands at constant pressure from 20 litres to 42 litres. Second the gas expands from 42 litres to 88 litres with a pressure drop according to the equation P = (100 - 0.8 V) kPa, where V is in litres. Calculate the work done on the gas. [Note that you need to calculate the initial pressure, which is not 100kPa.] a.-3889 J O b.-3669 J O c.-4199 J O d. -4039 J O e. 3539 J
The work done on the gas during the expansion process can be calculated by integrating the pressure with respect to the volume over each stage of the process. The total work done on the gas is approximately -3669 J.
To calculate the work done on the gas, we need to determine the pressure as a function of volume for each stage of the expansion process.
In the first stage, the gas expands at constant pressure. Since we know the initial and final volumes, we can calculate the constant pressure using the ideal gas law: PV = nRT. Given that the initial volume is 20 liters and the final volume is 42 liters, we have P₁ * 20 = nRT and P₂ * 42 = nRT, where P₁ and P₂ are the pressures at the initial and final states, respectively. Dividing the second equation by the first equation, we can solve for P₂/P₁ and find P₂ = 2.1P₁.
In the second stage, the pressure is given by the equation P = (100 - 0.8V) kPa. We can integrate this equation with respect to volume to find the work done during this stage.
The total work done on the gas is the sum of the work done in each stage. By integrating the pressure-volume relationship over each stage and summing the results, we find that the total work done on the gas is approximately -3669 J.
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Sue is on planet X which is a distance 6.6x109 km from a certain star which has a radius 7000 km. Sue measures the maximum intensity of light on the surface of the planet to be 9000 W m2. Planet X has no atmosphere and so there is no absorption of light between the star and the surface of the planet. Calculate the temperature of the star, which can be assumed to be a black body. a. 6.1e5K O b. 5.78 K O c. 2.0e5K O d. 6.0e6 K e. 6.0e8 K
Sue is on planet X which is a distance 6.6x109 km from a certain star which has a radius 7000 km. the correct option is (a) [tex]6.1 * 10^5 K.[/tex]
To calculate the temperature of the star, we can use the Stefan-Boltzmann Law, which states that the power radiated by a black body is proportional to the fourth power of its temperature (T):
Power = σ * A * T^4
Where:
Power is the total power radiated by the star
σ is the Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/(m^2·K^4))
A is the surface area of the star
First, we need to calculate the surface area of the star. Since it is a sphere, the surface area (A) is given by:
A = 4πr^2
Where r is the radius of the star (7000 km = 7 × 10^6 m).
A = 4π * (7 × 10^6)^2
A = 4π * 4.9 × 10^13
A ≈ 2.46 × 10^14 m^2
Now, we can rearrange the Stefan-Boltzmann Law to solve for T:
T^4 = Power / (σ * A)
Substituting the known values, including the power intensity (9000 W/m^2) measured by Sue on the planet's surface, we have:
T^4 = [tex]9000 W/m^2 / (5.67 * 10^{-8} W/(m^2.K^4) * 2.46 * 10^{14} m^2)[/tex]
T^4 ≈ [tex]6.48 * 10^{11} K^4[/tex]
Taking the fourth root of both sides:
T ≈ (6.48 × 10^11)^(1/4)
T ≈ 611,626 K
Rounding to the nearest hundredth, the temperature of the star is approximately [tex]6.1 * 10^5 K.[/tex]
Therefore, the correct option is (a)[tex]6.1 * 10^5 K.[/tex]
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A 4.00 kg particle is under the influence of a force F = 2y + x², where F is in Newtons and x and y are in meters. The particle travels from the origin to the coordinates (5,5) by traveling along three different paths. Calculate the work done on the particle by the force along the following paths. Remember that coordinates are in the form (x,y). a) In a straight line from the origin to (5,0), then, in a straight line from (5,0) to (5,5) b) In a straight line from the origin to (0,5), then, in a straight line from (0,5) to (5,5) c) In a straight line directly from the origin to (5,5) d) Is this a conservative force? Explain why it is or is not.
a) In a straight line from the origin to (5,0), then, in a straight line from (5,0) to (5,5)
The net work done by a force is given by:
Wnet = W1 + W2
Thus,W1 = ∫F . ds = ∫F (x)dx + ∫F (y)dy
Where,F (x) = 0F (y) = 2y + x²
∴ W1 = ∫(2y + x²) dy
= [y² + x²y]0 to 5
= (5² + 5²/2) − 0
= 25 + 12.5
= 37.5 J
Similarly,
W2 = ∫F (y)dy
= ∫(2y + x²)dy
= [y² + x²y]0 to 5
= (5² + 5²/2) − 0
= 25 + 12.5
= 37.5 J
Therefore, Wnet = 37.5 + 37.5 = 75 J
b) In a straight line from the origin to (0,5), then, in a straight line from (0,5) to (5,5)
W1 = ∫F (x)dx
= ∫(2y + x²) dx
= [2xy + x³/3]0 to 5
= (50 + 125/3) − 0
= 175/3 J
Similarly,
W2 = ∫F (y)dy = ∫(2y + x²)dy
= [y² + x²y]5 to 0
= (0 + 125/3) − 0
= 125/3 J
Therefore, Wnet = (175/3) + (125/3) = 100/3 J
c) In a straight line directly from the origin to (5,5)
W1 = ∫F . ds
= ∫F ds = ∫F dx + ∫F dy
F (x) = 2y + x²F (y) = 2y + x²
∴ W1 = ∫F (x) dx + ∫F (y) dy
= ∫(2y + x²) dx + ∫(2y + x²) dy
= [y² + x²y]0 to 5 + [y² + x²y]0 to 5
=37.5 J + 37.5 J= 75 J
D) Is this a conservative force? Explain why it is or is not.
The force is not conservative because the work done is different for the three different paths from the origin to (5, 5). In addition, the integral of the curl of the force is not equal to zero.
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Which one of the following is NOT equal to the potential energy stored on a fully charged capacitor with a capacitance of C farads, connected to a battery of V volts, and holding Q coulombs of charge? X A. (Q.V) joules A volt is a joule per coulomb, so multiplying volts by coulombs yields joules. X B. 1 2 (2) joules C A volt is a joule per coulomb, and capacitance is expressed as coulombs per volt, so dividing the square of coulombs by farads yields joules. A volt is a joule per coulomb, and capacitance is expressed as coulombs per volt, so multiplying farads by the square of voltage yields joules. 1/2 ( 7² ) joules volt is a joule per coulomb, and capacitance is expressed as coulombs per volt, so dividing the square of farads by volts yields coulombs to the fifth power divided by joules to the third power, not joules. X C. (C.V²) joules O D. 1 Detailed Guidance Send us feedback. Feedback Info Capacitors behave according to the equation: C = Q V where C is capacitance in farads, Q is charge in coulombs, and Vis volts. Since a volt is defined as one joule per coulomb, the charge leaving a discharging capacitor has energy of Q. Vcap joules. The voltage of a fully charged capacitor is equal to the voltage of the battery that charged it, but when the capacitor is almost completely discharged its voltage is essentially zero. Because of this, the actual energy stored on a capacitor is equal to (1/2)(Q Vbatt). Each of the answer choices is equivalent to this value except (D), which, because it does NOT represent the energy stored by the capacitor, is correct.
The potential energy stored on a fully charged capacitor with capacitance C farads, connected to a battery of V volts, and holding Q coulombs of charge is not equal to the expression given in option D.
Capacitors store energy in the form of electric potential energy. The energy stored on a capacitor can be calculated using the equation E = (1/2)(QV), where E is the energy in joules, Q is the charge in coulombs, and V is the voltage in volts. The voltage across the capacitor is equal to the voltage of the battery that charged it.
In the given options, option D states that the energy stored on the capacitor is 1 joule. However, this is incorrect. The correct expression for the energy stored on the capacitor is (1/2)(QV), which is equivalent to option A, B, and C. Option D does not represent the energy stored by the capacitor.
Therefore, the correct answer is option D. The potential energy stored on a fully charged capacitor with a capacitance of C farads, connected to a battery of V volts, and holding Q coulombs of charge is not equal to 1 joule.
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What is the net force acting on a 56 gram chicken egg that falls from a tree with a velocity of 5 m/s if it come to rest after 0.17 seconds?
Net force is the overall force that acts on an object. It is determined by adding up all of the individual forces acting on an object.
The net force acting on a 56-gram chicken egg that falls from a tree with a velocity of 5 m/s if it comes to rest after 0.17 seconds can be found as follows:
The mass of the chicken egg is 56 grams, and it can be converted to kilograms by dividing it by 1000.
56 g ÷ 1000 = 0.056 kg
The acceleration of the egg can be determined as
a = (v_f - v_i) / t where: v_f is the final velocity, v_i is the initial velocity, t is the time it takes to come to rest,
v_f = 0 (since the egg comes to rest)
v_i = 5 m/s
t = 0.17 s
a = (0 - 5 m/s) / 0.17 s⇒ a = -29.4 m/s²
To determine the net force acting on the egg, the formula for force can be used:
F = m × a
F = 0.056 kg × -29.4 m/s²
F = -1.6464 N
This gives the force that acted on the egg. The negative sign indicates that the force acted in the opposite direction to the velocity of the egg. However, the question asks for the net force, which means we have to take the magnitude of this value:
|F| = 1.6464 N
Thus, the net force acting on the egg is 1.6464 N.
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Disk 1 (of inertia m) slides with speed 4.0 m/s across a low-friction surface and collides with disk 2 (of inertia 2m) originally at rest. Disk 1 is observed to turn from its original line of motion by an angle of 15, while disk 2 moves away from the impact at an angle of 50°. Part A Calculate the final speed of disk 1. v1,f = _______ (Value) ________ (Units)
Part B Calculate the final speed of disk 2. v2,f = _______ (Value) ________ (Units)
Answer: Part A: v1,f = 2.31 m/s Part B: v2,f = 2.62 m/s
Part A Explanation
From the given problem, let's consider disk 1 slides with speed 4.0 m/s and the final velocity of disk 1 be v1,f.Now, the moment of inertia of disk 1 is m. From the principle of conservation of momentum and angular momentum, the following relation can be written:
mv1,i + 0 = mv1,f cos 15° + (mv1,f sin 15°)2mv1,
i = mv1,f cos 15° + (mv1,f sin 15°)2v1,
f = (2mv1,i)/(1.73 m)
Now, substituting the values, we get v1,
f = (2 x m x 4.0)/(1.73 x m) = 2.31 m/s.
Therefore, the final speed of disk 1 is v1,f = 2.31 m/s.
Part B Explanation
From the given problem, let's consider disk 2 with the final velocity v2,f and the moment of inertia 2m.From the principle of conservation of momentum and angular momentum, the following relation can be written.mv1,
i + 0 = 2mv2,f cos 50° + 0... (1)
Now, the impulse at the point of impact on disk 2 can be written as
f x t = (2mv2,f sin 50°)
(2)The vertical component of the equation
(2) can be used to find t as follows : f = m (v2,f - 0)/t => t = m (v2,f)/f.
Substituting t in equation (2) and simplifying, we get
v2,f = (mv1,i / 2m) (1/cos 50°)
Therefore, the final speed of disk 2 is v2,
f = (4.0 / 2) (1.31)
= 2.62 m/s.
Answer: Part A: v1,f = 2.31 m/s. Part B: v2,f = 2.62 m/s\
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The invisibility cloak from the Harry Potter books would be based on: An index of refraction that is exactly zero. An index of refraction that is between 0 and 1 An index of refraction that is greater than 2.5 A negative index of refraction
The invisibility cloak from the Harry Potter books would be based on a negative index of refraction.
In the Harry Potter books, the invisibility cloak allows the wearer to become completely invisible. Such an effect would require a material with unique optical properties. One possibility is a negative index of refraction.
In optics, the refractive index determines how light propagates through a medium. Normally, the refractive index of a material is positive, meaning light bends towards the normal when it enters the medium. However, a material with a negative refractive index would cause light to bend in the opposite direction, allowing it to curve around an object and effectively render it invisible. This concept is known as "invisibility cloaking" and has been a topic of scientific research. While achieving a true negative refractive index in practice is challenging, the invisibility cloak in the Harry Potter books is based on this idea, allowing the wearer to hide from view.
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An ultra-fast pulse lasers emits pulses of 13 fs. The length of each pulse train is: A) 7.79 pm B) 3.9 pm C) 19.49 pm D 11.69 pm ) E) 3.9 pm Air
An ultra-fast pulse lasers emits pulses of 13 fs. The length of each pulse train is: The correct answer would be that there is not enough information given to determine the length of each pulse train (option O).
To determine the length of each pulse train emitted by the ultra-fast pulse laser, we need to consider the relationship between the pulse duration and the pulse repetition rate.
The length of each pulse train is given by the formula:
Length of each pulse train = Pulse duration × Pulse repetition rate
The pulse duration is provided as 13 fs (femtoseconds). However, the pulse repetition rate is not given in the question. Without knowing the pulse repetition rate, we cannot accurately determine the length of each pulse train.
Therefore, based on the information provided, we cannot determine the exact length of each pulse train emitted by the ultra-fast pulse laser. The correct answer would be that there is not enough information given to determine the length of each pulse train (option O).
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A car travelling in a straight-line path has a velocity of +10.0 m/s at some instant. After 7.00 s, its velocity is +9.00 m/s. What is the average acceleration of the car during this time interval?
The average acceleration of the car during the given time interval is -0.14 m/s².
The given information are: Initial velocity (u) = +10.0 m/s Final velocity (v) = +9.00 m/s Time interval = 7.00 s. To calculate the average acceleration of a car during the given time interval, the formula is used below: Average acceleration, a = (v - u) / t Where, v is the final velocity, u is the initial velocity and t is the time interval. Substituting the given values: Average acceleration, a = (9.00 - 10.0) / 7.00a = -1.00 / 7.00
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How is the work done by the person related to the answers in parts A and B?
1. The work done by the person in lifting the book from the ground to the final height is the same as the answer to part A
2. The work done by the person in lifting the book from the ground to the final height is the same as the answer to part B
Neither statement accurately describes the relationship between the work done by the person and the answers in parts A and B.
The statement "The work done by the person in lifting the book from the ground to the final height is the same as the answer to part A" is incorrect. The work done by a person in lifting an object depends on the force applied and the distance over which the force is exerted, not solely on the height of the object.
Similarly, the statement "The work done by the person in lifting the book from the ground to the final height is the same as the answer to part B" is also incorrect. The work done in lifting the book is related to the change in potential energy, which depends on the mass of the book, the acceleration due to gravity, and the height difference between the initial and final positions. It is not directly related to the answer in part B.
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Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks)
Considering motion with a constant velocity, changes in distance are equal during equal time intervals. Since constant velocity is motion at a consistent speed in a straight line. It is possible to calculate the distance moved from the speed and the time taken.
Distance is equal to the product of speed and time: distance = speed × time. A constant speed in a straight line would result in a uniform change in distance for equal intervals of time.Considering motion with a non-constant velocity, changes in distance during equal time intervals are not equal. Since the velocity changes during non-constant velocity. Therefore the distance traveled in equal time periods will not be constant.
The object could be moving fast or slow, depending on the time interval you’re looking at. If the object's velocity is increasing, then the distance traveled in the same time interval will be greater.Speed is the rate at which an object travels from one place to another. It can be calculated by dividing distance by time.
In this case, speed = distance/time.100 meters in 15 seconds, speed = distance/time = 100/15 = 6.67 m/sIn 21 minutes, you ran 3000 meters east. To calculate the speed in km/min, convert the meters to kilometers and minutes to hours.
1 km = 1000 m and 1 hour = 60 minutes, therefore 3000 m = 3 km and 21 minutes = 21/60 = 0.35 hours.Speed = distance/time = 3/0.35 = 8.57 km/minVelocity is a vector quantity that indicates the rate and direction of an object's motion. An object moving at a constant speed in a straight line has constant velocity.
However, if an object is moving at a constant speed in a circular path, it is not moving at a constant velocity because its direction is constantly changing. For example, if a car is moving at 60 mph north, its velocity is 60 mph north. If it turns right, it's still moving at 60 mph, but its velocity is now 60 mph northeast.
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A 12.0 kg ladder leans against a frictionless wall. The ladder is 8.00 m long; it makes an angle of 52.0° with the floor. The coefficient of static friction between the floor and the ladder is 0.45. A 65.0 kg person is climbing the ladder. How far along the ladder can the person can climb before the ladder begins to slip? (a) Draw a diagram of the ladder depicting the forces acting on it. Clearly label each force. {Hint use descriptors such as mg, 0, etc.. not numerals} (b) Find how far along the ladder the person can climb before the ladder begins to slip.
(a) The free-body diagram of the ladder is shown below:1. Force of gravity on ladder = -mg (acts through the center of mass)2. Normal force from the floor on ladder = 0 (acts perpendicular to the floor and upward)3. Force of friction on ladder = -f_s (acts in a direction opposing motion)4. Force exerted by the person = P (acts parallel to the ladder and upward)5. Force of gravity on the person = -Mg (acts through the center of mass of the person)Free body diagram for ladder with forces acting on it.
(b) Calculate the maximum force of friction between the floor and ladder using the coefficient of static friction, 0.45, which is given by:f_s = μ_sN, where N is the normal force on the ladder from the floor. Since the ladder is not moving, the force of friction must be equal and opposite to the force exerted by the person on the ladder in order to maintain equilibrium:P = f_s = μ_sN, where N is the normal force on the ladder from the floor.
Therefore, the normal force is given by:N = Mg + m(gsinθ - μ_s cosθ), where θ is the angle the ladder makes with the floor. Substituting the given values, we get:N = (65.0 kg)(9.81 m/s^2) + (12.0 kg)(9.81 m/s^2)(sin 52.0° - 0.45 cos 52.0°)N = 772.2 NThe person can climb the ladder until the force exerted by the person on the ladder is equal to the maximum force of friction between the floor and ladder, which is:f_s = μ_sN = 0.45(772.2 N) = 347.5 NThe force exerted by the person is given by:P = Mg + mgsinθ = (65.0 kg)(9.81 m/s^2) + (12.0 kg)(9.81 m/s^2)(sin 52.0°)P = 784.4 N.
Therefore, the maximum distance along the ladder that the person can climb before the ladder begins to slip is given by:d = P/f_s = 784.4 N/347.5 N = 2.26 m (to three significant figures).Answer: (a) The free-body diagram of the ladder is shown above. (b) The maximum distance along the ladder that the person can climb before the ladder begins to slip is given by d = P/f_s = 2.26 m.
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The four drawings show portions of a long straight wire carrying current, I, in the presence of a uniform magnetic field directed into the page. In which case or cases does the wire feel a force to the left?
Using the right-hand rule, the direction of the force is downwards.Therefore, the wire will feel a force to the left in cases (a) and (c).
The given four drawings show portions of a long straight wire carrying current, I, in the presence of a uniform magnetic field directed into the page. In the cases, where the direction of the current and magnetic field are opposite to each other, the wire experiences a force to the left.In the given situation, the right-hand rule can be used to determine the direction of the force on a current-carrying wire in a magnetic field.
The rule states that if a right-handed screw is rotated in such a way that it moves in the direction of current and the magnetic field is represented by the direction of rotation of the screw, then the direction of force on the current-carrying wire will be in the direction of the screw that is pointing.The direction of force can be determined using Fleming's left-hand rule which states that if the thumb points in the direction of the current and the second finger in the direction of the magnetic field, then the direction of the force is perpendicular to both of them, which can be represented using the middle finger.
Using this rule, the following cases can be studied:Case (a): Here, the current flows upwards, and the magnetic field is directed into the page. Hence, using the right-hand rule, the direction of the force is towards the left.Case (b): In this case, the current flows downwards, and the magnetic field is directed into the page. Hence, using the right-hand rule, the direction of the force is towards the right.
Case (c): Here, the current flows from right to left, and the magnetic field is directed into the page. Hence, using the right-hand rule, the direction of the force is upwards.
Case (d): In this case, the current flows from left to right, and the magnetic field is directed into the page. Hence, using the right-hand rule, the direction of the force is downwards.Therefore, the wire will feel a force to the left in cases (a) and (c).
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A certain dense flint glass has an an index of refraction of nr = 1.71 for red light and nb = 1.8 for blue light. White light traveling in air is incident at an angle of 33.0° onto this glass. What is the angular spread between the red and blue light after entering the glass?
The angular spread between the red light and blue light after entering the glass is 0.8°.
The formula for angular dispersion is given as;
Δθ = θb - θr Where,
Δθ is the angular spread
θb is the angle of refraction for blue light
θr is the angle of refraction for red light
In this case, the angle of incidence is θi = 33.0°
Therefore,θi = θr (for red light)θi = θb (for blue light)
The formula for the angle of refraction is given as;
θ = arcsin(sin θi/n) Where,
θ is the angle of refraction
θi is the angle of incidence
n is the refractive index
On substituting the values given in the problem statement, we get;
For red light, θr = arcsin(sin 33.0°/1.71)
θr = 19.9°
For blue light,θb = arcsin(sin 33.0°/1.8)
θb = 19.1°
Therefore, the angular spread is;
Δθ = θb - θrΔθ = 19.1° - 19.9°Δθ = -0.8°
Thus, the angular spread between the red and blue light after entering the glass is -0.8°.
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Design FM transmitter block diagram for human voice signal with
available bandwidth of 10kHz
Also justify each block of your choice.
Design FM transmitter block diagram for human voice signal with
available bandwidth of 10kHz
The following are the justification for each block in block diagram of an FM transmitter for a human voice signal with an available bandwidth of 10 kHz:
Microphone: A microphone is a transducer that converts sound waves into electrical signals. As a result, the microphone should be of excellent quality, and the voice signal must be filtered and amplified to produce the necessary level of voltage.
Audio Amplifier: The audio signal that comes from the microphone has a very low level of voltage, therefore it must be amplified to increase the voltage to a level that is required for the modulator. As a result, the audio amplifier block must be included in the FM transmitter circuit.
RF Oscillator: The RF oscillator is the most important component of the FM transmitter. It produces a stable carrier signal that is modulated with the audio signal. A crystal-controlled oscillator is required for frequency stability.
Frequency multiplier: It is a multiplier circuit that increases the frequency of the carrier signal, which is necessary to get the desired output frequency. A frequency multiplier block must be included to achieve the desired output frequency.
Frequency Modulator: It is a circuit that modulates the audio signal onto the carrier signal. The frequency deviation is proportional to the amplitude of the audio signal. As a result, the frequency modulator block must be included in the FM transmitter circuit.
Power Amplifier: The power amplifier block is used to increase the power of the modulated signal to the level needed for transmission. As a result, it must be included in the FM transmitter circuit.
Antenna: It is the final stage of the FM transmitter. The modulated signal is transmitted by the antenna. Therefore, an antenna block is necessary to radiate the signal to the desired location.
This is the FM transmitter block diagram for a human voice signal with an available bandwidth of 10 kHz.
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An experimenter arranges to trigger two flashbulbs simultaneously, producing a big flash located at the origin of his reference frame and a small flash at x = 27.4 km. An observer, moving at a speed of 0.281c in the positive direction of x, also views the flashes. (a) What is the time interval between them according to her? (b) Which flash does she say occurs first?
(a) Number ___________ Units _______________
(b) __________
The time interval between the flashes according to the observer is 0.244 s.
The observer who is moving at a speed of 0.281c in the positive direction of x will say the flash occurs first.
(a) The distance between the flashes,
Δx = x2 – x1 = 27.4 km
The speed of light, c = 3 × 10^8 m/s
The speed of the observer, v = 0.281c
First, we need to calculate the Lorentz factor which is given by the formula;
γ = 1/√(1 - v²/c²)
γ = 1/√(1 - (0.281c)²/c²)
γ = 1/√(1 - 0.281²)
γ = 1.0481
Now, the time interval between the flashes according to the observer can be found out using the formula;
Δt' = γ Δt
Δt' = γ Δx/c
Δt' = (1.0481) (27.4 × 10³) / 3 × 10⁸
Δt' = 0.244 s
b) The observer who is moving at a speed of 0.281c in the positive direction of x would say that the small flash which is at x = 27.4 km occurs first.
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A 1.4 kg toy has an acceleration of 0.23 m/s2 when pushed with a force. A second toy has an acceleration of 0.75 m/s2 when pushed with the same force. What is the mass (in kg) of the second toy? Hint: Only enter the numerical value of your answer to two decimal places.
the required mass of the second toy is 0.43 kg.
The given force pushes a toy with a mass of 1.4 kg with an acceleration of 0.23 m/s². We are to calculate the mass of another toy that is pushed with the same force and has an acceleration of 0.75 m/s².We can use the following equation: force = mass × acceleration.
Therefore, we can write the following equations for the two toys:Force = (1.4 kg) × (0.23 m/s²)Force = mass × (0.75 m/s²)Solving the two equations for mass, we get:mass = Force/accelerationFor the first toy, we have:mass = (1.4 kg × 0.23 m/s²)/ (0.23 m/s²) = 1.4 kgFor the second toy, we have:mass = Force/acceleration = (1.4 kg × 0.23 m/s²)/ (0.75 m/s²) = 0.428 kgSo, the mass of the second toy is 0.43 kg (to two decimal places).Hence, the required mass of the second toy is 0.43 kg.
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Answer Both Parts Or Do Not Answer:
A uniform electric field is directed in the +x-direction and has a magnitude E. A mass 0.072 kg and charge +2.90 mC is suspended by a thread between the plates. The tension in the thread is 0.84 N.
What angle does the thread make with the vertical axis? Please give answer in degrees.
Find the magnitude of the electric force. Give answers in N to three significant figures.
The angle between the thread and the vertical axis is approximately 41.7 degrees. The magnitude of the electric force depends on the value of the electric field (E) and cannot be determined without that information.
To determine the angle the thread makes with the vertical axis, we can use trigonometry. The tension in the thread provides the vertical component of the force, and the electric force provides the horizontal component.
Given:
Mass (m) = 0.072 kg
Charge (q) = 2.90 mC = 2.90 × 10^(-3) C
Tension in the thread (T) = 0.84 N
The vertical component of the force is equal to the tension in the thread, so we have:
Tension (T) = mg
Solving for g, the acceleration due to gravity:
g = T / m
Substituting the values:
g = 0.84 N / 0.072 kg = 11.67 N/Kg
Next, we can find the magnitude of the electric force (F_e) using the formula:
F_e = qE
Given that the electric field magnitude (E) is directed in the +x-direction and has a value E, we can substitute the values:
F_e = (2.90 × 10^(-3) C) × E
The angle between the tension and the vertical axis can be found using the tangent function:
tan(theta) = Tension_y / Tension_x
tan(theta) = Weight / Tension
tan(theta) = 0.7056 N / 0.84 N
theta ≈ 41.7 degrees
Now, we can solve for θ by taking the inverse tangent (arctan) of both sides.
The magnitude of the electric force is given by F_e = (2.90 × 10^(-3) C) × E, where E is the electric field magnitude.
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Give your answers in SI units and to three significant figures. Question 1 3 pts Newer automobiles have filters that remove fine particles from exhaust gases. This is done by charging the particles and separating them with a strong electric field. Consider a positively charged particle +8μC that enters an electric field with strength 6×10 6
N/C. The particle is traveling at 77 m/s and has a mass of 1 g. If the horizontal width of the filter is 20 cm, determine the vertical distance that the particle will be deflected as it passes through the filter. Express your answer in meters.
The vertical distance that the particle will be deflected as it passes through the filter is 7.09 x 10^-6 m.
Explanation:Given,Charge of the particle, q = +8μC = +8 × 10^-6 CStrength of electric field, E = 6 × 10^6 N/CVelocity of the particle, v = 77 m/sMass of the particle, m = 1 g = 10^-3 kgWidth of the filter, d = 20 cm = 0.2 mThe electric force acting on a charged particle in an electric field is given byF = qE ……… (1)The particle will experience force in the horizontal direction, F = qE ……… (2)It will move with constant velocity in the vertical direction and experiences force of gravity in the vertical direction, F = mg ……… (3)Let ‘y’ be the vertical deflection. Net force experienced by the particle along the y-axis is given asFy = mg ……… (4)By Newton’s second law, F = ma ……… (5)Net force experienced by the particle along the x-axis is given asFx = qE ……… (6)Net force acting on the particle is given asFnet = √(Fx^2 + Fy^2) ……… (7)The net force acting on the particle is given asqE = ma ……… (8).
As the particle is moving with constant velocity along the y-axis, its acceleration along the y-axis is zero.Therefore, Fy = 0mg = 0y = 0Also, the net force acting on the particle is given by, Fnet = qE ……… (9)Fnet = qE = +8 × 10^-6 × 6 × 10^6 = 48 × 10^-6 NNet force acting on the particle along the x-axis is given as,Fx = Fnet sin θ ……… (10)θ = tan^-1 (y/d)Fx = ma = Fnet cos θ ……… (11)θ = tan^-1 (y/d)a = Fnet/m = (qE)/mcos θsin θ = y/dcos θ = √(1 – sin^2 θ)cos θ = √(1 – (y/d)^2)Fx = ma = Fnet cos θ(8 × 10^-6) × 6 × 10^6 √(1 – (y/0.2)^2) = (10^-3) × ay/0.2 = (48 × 10^-6)/[(10^-3) × 6 × 10^6 √(1 – (y/0.2)^2)]y = 7.09 x 10^-6 m.
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