The Electric Potential Existing in Space is V ( X , Y , Z ) = a ( X Y + Y Z + Z X ) . (A) Write the Dimensional Formula of A.(B) Find the Expression for the Electric Field. - Physics

Numerical

The electric potential existing in space is $\hspace{0.167em} V(x, y, z) = A(xy + yz + zx) .$ (a) Write the dimensional formula of A. (b) Find the expression for the electric field. (c) If A is 10 SI units, find the magnitude of the electric field at (1 m, 1 m, 1 m).

Solution

Given:
Electric potential,

$V(x, y, z) = A(xy + yz + zx)$

$A = \frac{\text{ volt }}{m^2}$

$\Rightarrow \left[ A \right] = \frac{\left[ {ML}^2 I^{- 1} T^{- 3} \right]}{\left[ L^2 \right]}$

$\Rightarrow A = [ {MT}^{- 3} I^{- 1} ]$

(b) Let be the electric field.

$dV = - \vec{E} . \vec{dr}$

$\Rightarrow A(y + z)dx + A(z + x)dy + A(x + y)dz = - E(dx \hat{i} + dy \hat{j} + dz\hat{ k } )$

$\Rightarrow [A(y + z) \hat{i } + A(z + x)\hat{ j } + A(x + y) \hat{ k } ] [dx\hat{ i} + dy \hat{j } + dz \hat{k } ] = - E\left[ dx \hat{ i }+ dy\hat{ j } + dz \hat{ k } \right]$

Equating now, we get

$\vec{E} = - A(y + z) \hat{ i } - A(z + x) \hat{ j } - A(x + y) \hat{ k }$

(c) Given: A = 10 V/m2

$r = (1 m, 1 m, 1 m)$

$\vec{E} = - 10 (2) \hat{ i } - 10 (2) \hat{ j } - 10 (2) \hat{ k }$

$= - 20 \hat{ i } - 20 \hat{ j } - 20 \hat{ k }$

Magnitude of electric field,

$\left| E \right| = \sqrt{{20}^2 + {20}^2 + {20}^2}$

$= \sqrt{1200} = 34 . 64 = 35$ N/C

Concept: Electric Field - Electric Field Due to a System of Charges
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 7 Electric Field and Potential
Q 60 | Page 123