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BITSAT 2010
Exam
BITSAT
Year
2010
Subject
Physics
Chapter
Electrostatics

BITSAT 2010 · Physics · Electrostatics

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

In a region, the potential is represented by V(x,y,z)=6x−8xy−8y+6yz,\mathrm{V}(\mathrm{x}, \mathrm{y}, \mathrm{z})=6 \mathrm{x}-8 \mathrm{xy}-8 \mathrm{y}+6 \mathrm{yz}, where V\mathrm{V} is in volts and x,y,z\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 :

Options

  1. A65N6 \sqrt{5} \mathrm{ N}
  2. B30N30 \mathrm{ N}
  3. C24N24 \mathrm{ N}
  4. D435N4 \sqrt{35} \mathrm{ N}

Correct answer

D. 435N4 \sqrt{35} \mathrm{ N}

Step-by-step solution

E→=−∂V∂xi^−∂V∂yj^−∂V∂zk^=−[(6−8y)i^+(−8x−8+6z)j^+(6y)k^]At(1,1,1),E→=2i^+10j^−6k^⇒(E→)=22+102+62=140=235∴\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}E→=2×235=435\overrightarrow{\mathrm{E}}=2 \times 2 \sqrt{35}=4 \sqrt{35}
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