Question

In a region, the potential is represented by V(x, y, z) = 6x - 8xy - 8y + 6yz, where $V$ is in volts and $x, y, z$ are in metres. The electric force experienced by a charge of $2$ coulomb situated at point $(1, 1, 1)$ is

• A. $ 6 \sqrt 5 \,N $
• B. 30 N
• C. 24 N
• D. $ 4 \sqrt 35 \,N $

Answer 1

Answer: (D)

Here, V(x, y, z) = 6x - 8xy - 8y + 6yz
The x, y and z components of electric field are
$ E_x = - \frac{\partial V}{\partial x} = - \frac{\partial}{\partial x} (6x - 8xy - 8y + 6yz) $
$ \, \, \, \, \, \, = -(6 - 8y) = -6 + 8y $
$ E_y = - \frac{\partial V}{\partial y} = - \frac{\partial}{\partial y} (6x - 8xy - 8y + 6yz) $
$ \, \, \, \, \, \, \, = -(-8x - 8 + 6z) = 8x + 8 - 6z $
$ E_z = - \frac{\partial V}{\partial z} = - \frac{\partial}{\partial z} (6x - 8xy - 8y + 6yz) = - 6y $
$ \overrightarrow {E} = E_x \widehat {i} + E_y \widehat {j} + E_z \widehat {k} $
$ = (-6 + 8y) \widehat {i} + (8x + 8 - 6z) \widehat{j} - 6y \widehat {k} $
At point (1, 1, 1)
$ \overrightarrow {E} = (-6 + 8) \widehat {i} + (8 + 8 - 6) \widehat {j} - 6 \widehat {k} $
$ = 2 \widehat {i} + 10 \widehat {j} - 6 \widehat {k} $
The magnitude of electric field $ \overrightarrow {E} $ is
$ \overrightarrow {E} = \sqrt {E_x^2 + E_y^2 + E_z^2} = \sqrt {(2)^2 + (10)^2 + (-6)^2} $
$ = \sqrt 140 = 2 \sqrt 35\, N\, C^{-1} $
Electric force experienced by the charge
$ F = qE = 2 \,C \times 2 \sqrt 35\, N \,C^{-1} = 4 \sqrt 35\, N $