Question

HOMERWORK SET1-Electrostatics Due Date Thu, Sept 20th fv-22y2 V in free space, fnd the eergy stored in a lme defined by 1 sI, Hint: Given V(x.y). we can get the eectric field since E-grad(V) A spherical conductor ofradíus α carries a surface charge with density pa-Determine the potential energy in terms of a. 2. 3 IfE-3,5a V/m, calculate the potential energy stored within the vokume defined by o r< 1,0<y<2,0fc3 4. In free space, Vpe sinip) (a) find E (b) Determine the energy stored in the region 0 кре 1,0 <p Given that D-50a, + 80ay-Xla, nC/m, n regon x-0 where 1-2. lio, find D in region x-c) where &7.6 A conducting sphere of radius a is half embecded in a liquid dielectric medium of permittivity a. The region above the liquid is a gas of permittivity ez. If the total free charge on the sphere is Q, determine the electric field intensity everywhere. 2s, О 2. 5. 6. 7. (a) Given that E 15a, 8a, Vm at a point on a conductor surface, what is the surface charge density at that point? Assume- (b) Region y 2 is occupied by a conductor. Ifthe surface charge on the conductor is -20 nCm, find D just outside the conductor. A spherical capacitor has inner radius d and outer radius a. Concentric with the spherical conductors and lying between them is a spherical shell of outer radius e and inner radius b. If the regionsdsrse,cb, andbraare filled with materials with permittivities si ez, and e, respectively, determine the capacitance on the system 8.

Answer 1

1. V = 2x^2 + 6y^2 V

-1 <= x <= 1

-1 <= y <= 1

-1 <= z <= 1

E = -grad(V) = -4xi - 12yj

energy stored inelectric field = U

U = Volume*epsilon*E^2/2

dU = epsilon(16x^2 + 144y^2)*dxdydx/2

dU = epsilon(16x^2 + 144y^2)dxdy

dU = epsilon(32/3 + 288y^2)dy

dU = epsilon(32*2/3 + 288*(2/3))

U = 1.8904818*10^-9 J

2. spherical conductor radius a

surface charge density, rho

potential energy U(r)

0 < r < a

V(r) = k*rho*4*pi*a^2/a = 4*pi*k*rho*a

E(r) = 0

hence

U(r) = 0

for

a < = r

V(r) = 4*pi*k*rho*a^2/r

E(r) = 4*pi*k*rho*a^2/r^2

dU = E(r)^2*epsilon*4*pi*r^2*dr

integrating from r = a to r = r

U = 16*pi^2*k^2*rho^2*a^4*epsilon*4*pi(1/a - 1/r)

U = rho^2*a^4*4*pi(1/a - 1/r)/epsilon