V. [25 points total] This page contains two independent parts, A and B A. The el

Question

V. [25 points total] This page contains two independent parts, A and B A. The electric field of a dipole is shown in the vector field diagram at right. The dipole consists of a single positive point charge above and a single negative point charge below. [5 pts] At point D, is the divergence of the electric field positive, negative, or zero? Explain your reasoning. i. Vector field diagram (not a field line diagram [5 pts] At point C, in what direction is the curl of the electric field? If the curl of the electric field is zero, state so explicitly. Explain your reasoning ii. B. A circular disk of radius R: centered at the origin and oriented in the xy-plane is charged with a surface charge density of (xg.z) as. Point P is located a distance / above the origin along the z-axis, at coordinates (0, 0, /). Point O is located on the disk a distance Ri from the origin, at coordinates (Ri, 0, 0). i. Consider a small region of the disk around point O [8 pts] How much charge is located in that small region? Use an appropriate coordinate system and variables provided above. Show your work and/or explain your reasoning. Perspective view 13 pts] What is the distance from the coordinate (Ri, 0, 0) to point P? Show your work. The electric potential at a point relative to infinity is V = kq/d for a point charge, where d is the distance between the point charge and where the electric potential is measured, and k is a constant. [4 pts] Based on your answers above, write an integral expression for the electric potential (not electric field) at point P due to the charged disk, relative to infinity. You do not need to evaluato the integral. ii.

Explanation / Answer

A. 1. For the given dipole

let distance between the charges be a, distance of point D from +q be x

then electric field at D is kq[1/x^2 - 1/(x + a)^2]

now, divergence of E is dE/dx

D.E = dE/dx = 2kq[-1/x^3 + 1/(x + a)^3]

as x + a > x

D.E < 0, hence the divergence at point D will be -ve

2. at point C, electric field is along the direction antiparallel to the direction of the dipole moment, direction -x

then let distance of the point C form the midpoint of the dipole be y

then E = -kqa/(y^2 + a^2/4)^3/2 i ( where i ias a unit vector along x axis)

so curl of E wil be

CxE = -dE/dy k = -3kqay/(y^2 + a^2/4)^5/2

hence cul of the electric field is -ve

B. given radius of disc = R2

surface charge densityt sigma = ao*s ( where s is the distance form the origin)

1. consider point Q

s = R1

hence

sigma = ao*R1

hence charge in small area dA around point Q will be dQ = sigma*dA

dQ = ao*R1*dA

2. coordinates of point P = (0,0,l)

coordinates of point X = (R1,0,0)

hence distance between these two points = sqroot(R1^2 + l^2)

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