Question

The electric potential in a region is given by V(x,y,z) = -10.0x2 + 20.0xyz + 6.0y3
a) Find the electric field that produces this potential?
b) Find the amount of charge contained within a cubic region in space 20 cm on a side and centered at the point (10.0 cm, 10.0 cm, 10.0 cm).

Answer 1

(a) here potential V(x,y,z) = -10.0 x² + 20.0 xyz + 6.0 y³

so the electric field is given by

E = -( dV/dx + dV/dy + dV/dz )

= - [ (-20x +20yz) + ( 20xz + 18y² ) + (20xy) ]

= - [ -20x +18y² +20xy +20yz +20xz ] V/m Ans

(b) as we know that the electric field

E = kq / r²

here r = 20cm = 0.2 m and k = 9 x 10⁹

- [ 20x + 18y² + 20xy + 20yz + 20xz ] = ( 9 x 10⁹ ) q / (0.2)²

here x = y = z = 10cm = 0.1 m

so - [ 2 + 0.18 + 0.2 + 0.2 + 0.2 ] = ( 225 x 10⁹ )q

q = - 2.78 / (225 x 10⁹)

= - 0.01235 x 10^-9 Cb Ans