A spherical cloud of total charge + Q has radius R. The charge density of this c

Question

A spherical cloud of total charge + Q has radius R. The charge density of this cloud is non-uniform varying according to Where r is the distance from the center of the cloud. Express all allgebraic answers in terms of Q,R and fundamental constants. (A) Determine the following as a function of r for r > R o The magnitude E of the electric field, o The electric potential V. (B) A proton is placed at point P shown and released. Describe completely its motion for a long time after its release D) An electron of charge magnitude e is now placed at point P. which is a distance r from the center of the sphere and released. Determine the kinetic energy of the electron as a function of r as it strikes the cloud. (D) Derive an expression for rho 0. (E) Determine the magnitude of the electric field as a function of r for r lessthanorequalto R

Explanation / Answer

part a )

from gauss law = E.dA = Q/eo

E = Q/eo*4pi*r^2

V = kQ/r

part b )

speed will increase acceleratio will decrease

part c )

KE = change in work

KE = integral F.dr from r ot R = -KQqintegraldr/r^2 from r to R

KE = KQq*(1/R - 1/r )

part d )

Q = integral rho*dV from 0 to R

Q = integral rho_0(1-r/R) *4pi*r^2dr from 0 to R

Q =4pi*rho_0 integral (1-r/R)r^2dr = 4pi*rho_0 [ r^3/3 - r^4/4R] from 0 to R

= 4pi*rho_0* R^3 [ 1/3 - 1/4] = 4pi*rho_0 *R^3/12 = pi*rho_0*R^3 /3

rho_0 = 3Q/pi*R^3

part e )

E.dA = Q/eo

E.4pi/r^2 = 1/eo * integral rho*dV = 1/eo * integral rho_0(1-r/R)*4pi*r^2*dr

= 4pi*rho_0/eo * (r^3/3 - r^4/4R) from 0 to r

4pi*rho*r^3/eo * [ 1/3 - r/4R]

E = kQr/R^3 *[ 4 -3r/R]

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