Answer to Question #213499 in Electricity and Magnetism for Muthiah Afifah

Question

A potential field is given by V 5 3x2 y 2 yz. Which of the following is not true?

(a) At point 11, 0, 212, V and E vanish.

(b) x2 y 5 1 is an equipotential line on the xy-plane.

(c) The equipotential surface V 5 28 passes through point P+2, -1, 4

. (d) The electric field at P is 12ax 2 8ay 2 az V/m.

(e) A unit normal to the equipotential surface V 5 28 at P is 20.83ax 1 0.55ay1 0.07az.

Answer 1

$V=3x^2y^2-yz$

$E=-\nabla V=-\nabla(3x^2y^2-yz)$

$$E=-(6xy^2a_x+(6x^2y-z)a_y-ya_z)$$

$E_{(11,0,212)}=-0a_x+212a_y+0a_z$

$V_{11,0,212}=0$

Option (a) incorrect

Part(b)

$P=x^2y-1$

$\nabla.P=2xy\rightarrow(1)$

equation (1) lies x-y plane

Option (b) is correct option

Part(c)

V=28

$V_{2,-1,4}=16$

Option (c) incorrect

Part(d)

Electric field

$$E=-(6xy^2a_x+(6x^2y-z)a_y-ya_z)$$

$$E_{2,-1,4}=-(6\times(-1)\times16a_x+(6\times4\times(-1)-4)a_y+1\times a_z$$

$$E_2,-1,4=-(-96a_x-28a_y+a_z)$$

$E_{2,-1,4}=96a_x+28a_y-a_z$

Option (d) is incorrect

Part(e)

$\hat{V}=\frac{V}{|V|}$

$\hat{V}=\frac{-(6xy^2a_x+(6x^2y-z)a_y-ya_z}{30.48}$

$\hat{V}=\frac{-(12a_x-28a_y+a_z)}{30.47}$

$\hat{V}=-0.39a_x+0.91a_y-0.032a_z$

Option (e) is incorrect