The electric potentioal V in a region of space is given by V(x,y,z)=A(x 2 -3y 2

Question

The electric potentioal V in a region of space is given by V(x,y,z)=A(x2-3y2+z2) where A is a constant.

(b) The work done by the field when a 1.50-uC test charge moves from the point (x,y,z)=(0,0,0.250m) to the origin is measured to be 6.00 X 10-5J.Determine A.

(c) Determine the electric field at the point (0,0,0.250m).

(d) Show that in every plane parallel to the xz-plane the equipotential contours are circles.

(e) What is the radius of the equipotential contour corresponding to V = 1280V and y = 2.00m?

Explanation / Answer

V = Ax^2 - 3Ay^2 + Az^2

a) E = - dV/dr

Ex = -dV/dx = -2Ax

Ey = - dV/dy = 6Ay

Ez = -2Az

E = -2Ax i + 6Ay j - 2Az k

b) change in potential =A [ (0^2 - 0 + 0.250^2)] = 0.0625A

work done = q deltaV

6 x 10^-5 = 1.50 x 10^-6 x 0.0625A

A = 640 V / m^2

c) E = 0i + 0j + (-2 x 640 x 0.250)k

E = - 320k N/C

d) xz plane consists of x and z point (variable)

and there will be some value of y (fixed for that plane)

so V/A = x^2 - 3(y1)^2 + z^2

x^2 + z^2 = (V/A + 3y1^2 ) = constant

this is the equation of circle.

with centre origin and

radius = sqrt[ (V/A + 3y1^2 ) ]

e) V/A = 1280 / 640 = 2

3y^2 = 3(2^2) = 12

radius = sqrt(2+12) = 3.74 m

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