A thin spherical spherical shell of radius R which carried a uniform surface charge density σ. Write an expression for the volume charge density rho(r) of this thin shell distribution in terms of σ and an appropriate delta function. Verify explicitly that the units of your final expression are correct. Also show that your total integrated charge comes out right.

Answer:

Explanation:

From the given information:

We know that the thin spherical shell is on a uniform surface which implies that both the inside and outside the charge of the sphere are equal, Then

The volume charge distribution relates to the radial direction at r = R

∴

[tex]\rho (r) \ \alpha \ \delta (r -R)[/tex]

[tex]\rho (r) = k \ \delta (r -R) \ \ at \ \ (r = R)[/tex]

[tex]\rho (r) = 0\ \ since \ r< R \ \ or \ \ r>R---- (1)[/tex]

To find the constant k, we  examine the total charge Q which is:

[tex]Q = \int \rho (r) \ dV = \int \sigma \times dA[/tex]

[tex]Q = \int \rho (r) \ dV = \sigma \times4 \pi R^2[/tex]

∴

[tex]\int ^{2 \pi}_{0} \int ^{\pi}_{0} \int ^{R}_{0} \rho (r) r^2sin \theta \ dr \ d\theta \ d\phi = \sigma \times 4 \pi R^2[/tex]

[tex]\int^{2 \pi}_{0} d \phi* \int ^{\pi}_{0} \ sin \theta d \theta * \int ^{R}_{0} k \delta (r -R) * r^2dr = \sigma \times 4 \pi R^2[/tex]

[tex](2 \pi)(2) * \int ^{R}_{0} k \delta (r -R) * r^2dr = \sigma \times 4 \pi R^2[/tex]

Thus;

[tex]k * 4 \pi \int ^{R}_{0} \delta (r -R) * r^2dr = \sigma \times R^2[/tex]

[tex]k * \int ^{R}_{0} \delta (r -R) r^2dr = \sigma \times R^2[/tex]

[tex]k * R^2= \sigma \times R^2[/tex]

[tex]k = R^2 --- (2)[/tex]

Hence, from equation (1), if k = [tex]\sigma[/tex]

[tex]\mathbf{\rho (r) = \delta* \delta (r -R) \ \ at \ \ (r=R)}[/tex]

[tex]\mathbf{\rho (r) =0 \ \ at \ \ r<R \ \ or \ \ r>R}[/tex]

To verify the units:

[tex]\mathbf{\rho (r) =\sigma \ * \ \delta (r-R)}[/tex]

↓         ↓            ↓

c/m³    c/m³  ×   1/m            

Thus, the units are verified.

The integrated charge Q

[tex]Q = \int \rho (r) \ dV \\ \\ Q = \int ^{2 \ \pi}_{0} \int ^{\pi}_{0} \int ^R_0 \rho (r) \ \ r^2 \ \ sin \theta \ dr \ d\theta \ d \phi \\ \\ Q = \int ^{2 \pi}_{0} \ d \phi \int ^{\pi}_{0} \ sin \theta \int ^R_{0} \rho (r) r^2 \ dr[/tex]

[tex]Q = (2 \pi) (2) \int ^R_0 \sigma * \delta (r-R) r^2 \ dr[/tex]

[tex]Q = 4 \pi \sigma \int ^R_0 * \delta (r-R) r^2 \ dr[/tex]

[tex]Q = 4 \pi \sigma *R^2[/tex]    since  [tex]( \int ^{xo}_{0} (x -x_o) f(x) \ dx = f(x_o) )[/tex]

[tex]\mathbf{Q = 4 \pi R^2 \sigma }[/tex]

What is the displacement of the object?

A sprinter begins from rest and accelerates at the rate of 2 m/s^2 for 200 m.
a. What is the sprinters velocity at the end of the 200 m?
b. How long does it take him to cover it?
c. What is his average velocity?

Upon impact, bicycle helmets compress, thus lowering the potentially dangerous acceleration experienced by the head. A new kind of helmet uses an airbag that deploys from a pouch worn around the rider's neck. In tests, a headform wearing the inflated airbag is dropped from rest onto a rigid platform; the speed just before impact is 6.00 m/s. Upon impact, the bag compresses its full 12.0 thickness, slowing the headform to rest.

Required:
Determine the acceleration of the head during this event, assuming it moves the entire 12.0 cm.

___ is the pull that all objects exert on each other.
A.Resistance
B.Inertia
C.Gravity
D. Force