Question

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).

Answer 1

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 E 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