In a region, the potential is represented by V(x, y, z)=6 x-8 xy-8 y+6 yz, 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 :

Correct answer: (D) 4 √35 ~N. overrightarrowE=-(∂ V)/(∂ x) i-(∂ V)/(∂ y) j-(∂ V)/(∂ z) k =-[(6-8 y) i+(-8 x-8+6 z) j+(6 y) k] At(1,1,1), overrightarrowE=2 i+10 j-6 k ⇒ (overrightarrowE)=√2^2+10^2+6^2=√140=2 √35 ∴ F=q overrightarrowE=2 × 2 √35=4 √35

Q111 / 125 · BITSAT 2010 · Physics

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

A
$6 \sqrt{5} \mathrm{~N}$
B
$30 \mathrm{~N}$
C
$24 \mathrm{~N}$
Correct answer
D
$4 \sqrt{35} \mathrm{~N}$

Solution

$\overrightarrow{\mathrm{E}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}} \hat{\mathrm{i}}-\frac{\partial \mathrm{V}}{\partial \mathrm{y}} \hat{\mathrm{j}}-\frac{\partial \mathrm{V}}{\partial \mathrm{z}} \hat{\mathrm{k}}$ $=-[(6-8 \mathrm{y}) \hat{\mathrm{i}}+(-8 \mathrm{x}-8+6 \mathrm{z}) \hat{\mathrm{j}}+(6 \mathrm{y}) \hat{\mathrm{k}}]$ $\mathrm{At}(1,1,1), \overrightarrow{\mathrm{E}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ $\Rightarrow \quad(\overrightarrow{\mathrm{E}})=\sqrt{2^{2}+10^{2}+6^{2}}=\sqrt{140}=2 \sqrt{35}$ $\therefore \quad \mathrm{F}=\mathrm{q} \overrightarrow{\mathrm{E}}=2 \times 2 \sqrt{35}=4 \sqrt{35}$

Hence, the correct answer is option: 'D'.

Solution

Relevant Questions