The electric potential in a volume of space is given by V(x, y, z) = 4x^3 + x^2y

Question

The electric potential in a volume of space is given by V(x, y, z) = 4x^3 + x^2yz^3 + 5y^2z^2. Determine the electric field in this region at the coordinate (3, 4, 6). A conducting solid sphere (radius of R = 18.0 cm. charge of q = 6.10 10-6 C) is shown in the figure. Calculate the electric potential at a point 24.0 cm from the center, a point on the surface, and at the center of the sphere. Assume that the electric potential is zero at points infinitely far away from the origin of the coordinate system.

Explanation / Answer

3) elctric potential V = 4x^3 +x^2yz^3 +5y^2z^2

Electric filed E = - dV/dr

E = -dV/dx i -dV/dy j -dV/dz

E = -(12x^2 +2xyz^3) i -(x^2z^3 +10yz^2)j -(3x^2yz^2 +10y^2z) k

r = (x,yz) = (3,4,6)

E = -(12*3^2 +2*3*4*6^3) i -(3^2*6^3 +10*4*6^2)j - (3*3^2*4*6^2 +10*4^2*6) k

E = -5292 i - 3384 j -4848 k

4) q = 6*10^-6 X, R = 18 cm

(A) r = 24 cm

r >R

V = kq/r

V = 9*10^9*6*10^-6/0.24

V = 225000 V

(B) R = 18 cm

V = 9*10^9*6*10^-6/0.18

V = 300000 V

(c) for conducting sphere, it is equipotential surface

for r < R

Vr = VR

V = 300000 V

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