Exercise 23.45 In a certain region of space, the electric potential is V ( x , y

Question

Exercise 23.45

In a certain region of space, the electric potential is V(x,y,z)=AxyBx2+Cy, where A, B, and C are positive constants.

Part A

Calculate the x-component of the electric field.

Express your answer in terms of the given quantities.

Part B

Calculate the y-component of the electric field.

Express your answer in terms of the given quantities.

Part C

Calculate the z-component of the electric field.

Express your answer in terms of the given quantities.

Part D

At which points is the electric field equal to zero?

At which points is the electric field equal to zero?

1- x=0,y=0,z=0 2- x=C/A,y=0,z=2BC/A2 3- x=C/A,y=2BC/A2, any value of z 4- x=2BC/A2, any value of y,z=C/A 5- There is no point at which the electric field equal to zero.

Explanation / Answer

here,
according to Question:

V(x,y,z) = A*x*y B*x^2 + C*y

as we know

E = - grad(V)

Where grad is gradient operator. which is just for taking derivative with respect to its variable x,y and z.

Part A:
Ex = - dV/dX = -Ay + 2Bx

PartB:
Ey = - dV/dy = -Ax - C

Part C:
Ez = - dV/dz = 0

Part D:

Field Will be Zero when both Ex and Ey will be equal to zero.

Ex = 0
-Ay + 2Bx = 0
y = 2Bx/A -----------------------(1)

also

Ey = 0
-Ax - C = 0
x = C/A--------------------(2)

using 2 in 1 , we get

y = 2BC/A^2

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