1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ne4ueva [31]
3 years ago
12

A thin spherical spherical shell of radius R which carried a uniform surface charge density σ. Write an expression for the volum

e 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.
Physics
1 answer:
ozzi3 years ago
6 0

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

∴

\rho (r) \  \alpha  \  \delta (r -R)

\rho (r) = k \  \delta (r -R) \ \  at \ \  (r = R)

\rho (r) = 0\ \ since \ r< R  \ \ or  \ \ r>R---- (1)

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

Q = \int \rho (r) \ dV = \int \sigma \times dA

Q = \int \rho (r) \ dV = \sigma \times4 \pi R^2

∴

\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

\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

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

Thus;

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

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

k * R^2= \sigma \times  R^2

k  =   R^2 --- (2)

Hence, from equation (1), if k = \sigma

\mathbf{\rho (r) = \delta* \delta (r -R)  \ \  at   \ \  (r=R)}

\mathbf{\rho (r) =0 \ \  at   \ \  rR}

To verify the units:

\mathbf{\rho (r) =\sigma \ *  \ \delta (r-R)}

↓         ↓            ↓

c/m³    c/m³  ×   1/m            

Thus, the units are verified.

The integrated charge Q

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

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

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

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

\mathbf{Q = 4 \pi R^2  \sigma  }

You might be interested in
How do you calculate energy lost due to friction in an experiment?
Snowcat [4.5K]

Answer:

A treadmill get it? but its   Ff * d cos theta

Explanation:

6 0
3 years ago
Write the first equation of motion. Under what condition(s) is this equation valid?​
Zepler [3.9K]

Explanation:

The first equation of motion in kinematics is given by :

v=u+at .....(1)

u is initial speed

a is acceleration

v is final speed

t is time

Equation (1) is valid when the object is moving with constant acceleration. This equation gives relation between velocity and time.

3 0
3 years ago
What is the kinetic energy of a 55 kilogram skier traveling at meters per second
Rasek [7]
How many meters per second was it traveling
4 0
3 years ago
A difference In density between gases creates buoyancy which is the ability of an object to float. 0°C helium has a density of .
rjkz [21]

Answer:

mass = 0.18 [kg]

Explanation:

This is a classic problem where we can apply the definition of density which is equal to mass over volume.

density = \frac{mass}{volume} \\\\where:\\volume = 1 [m^3]\\density = 0.18[kg/m^3]

mass = 0.18*1

mass = 0.18 [kg]

6 0
3 years ago
1. What types of gas was the first atmosphere made out of?
sergij07 [2.7K]
17. a sea breeze is a breeze blowing towards the land
18. a land breeze is a breeze blowing towards the sea
8 0
3 years ago
Read 2 more answers
Other questions:
  • An _____ contains in maximum number of hydrogen atoms possible
    10·1 answer
  • Two balls collide with each other. Ball 1 experiences a high acceleration as a result of the collision, while Ball 2 experiences
    7·2 answers
  • Which atomic model was proposed as a result of j. J. Thomson’s work?
    13·1 answer
  • The Hubble telescope’s orbit is 5.6 × 105 meters above Earth’s surface. The telescope has a mass of 1.1 × 104 kilograms. Earth e
    10·1 answer
  • What is the speed of a runner that runs 400m in 48.42 seconds?
    10·1 answer
  • does Bart Allen from the young justice show like men cause some one told me that watching season 3 and is the flash a bad guy fo
    12·1 answer
  • A block slides down a frictionless plane that makes an angle of 24.0° with the horizontal. What is the
    14·1 answer
  • Tasks
    7·1 answer
  • I cant solve this problem, and our teacher said that this would be in the test we'll have tomorrow, can someone help me?
    14·1 answer
  • what seems to happen to the one electron in the outer layer of sodium when it combines with chlorine to form sodium chloride?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!