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Concepts Of Physics MCQ Edition [Volume 2]
Exam
Concepts Of Physics MCQ Edition [Volume 2]
Subject
Physics
Chapter
Electric Field and Potential

Concepts Of Physics MCQ Edition [Volume 2] · Physics · Electric Field and Potential

The electric potential in a region of space is described by V(x, y,...

The electric potential in a region of space is described by V(x,y,z)=A(xy+yz+zx)V(x, y, z) = A(xy + yz + zx). Find the expression for the electric field.

Options

  1. A−A{i^(x)+j^(y)+k^(z)}-A\{\hat{i}(x) + \hat{j}(y) + \hat{k}(z)\}
  2. B−A{i^(y+z)+j^(z+x)+k^(x+y)}-A\{\hat{i}(y + z) + \hat{j}(z + x) + \hat{k}(x + y)\}
  3. CA{i^(y+z)+j^(z+x)+k^(x+y)}A\{\hat{i}(y + z) + \hat{j}(z + x) + \hat{k}(x + y)\}
  4. D−A{i^(yz)+j^(zx)+k^(xy)}-A\{\hat{i}(yz) + \hat{j}(zx) + \hat{k}(xy)\}

Correct answer

B. −A{i^(y+z)+j^(z+x)+k^(x+y)}-A\{\hat{i}(y + z) + \hat{j}(z + x) + \hat{k}(x + y)\}

Step-by-step solution

The electric field E⃗\vec{E} is related to the electric potential VV by the negative gradient of the potential:

E⃗=−∇V=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\vec{E} = -\nabla V = -\left( \dfrac{\partial V}{\partial x}\hat{i} + \dfrac{\partial V}{\partial y}\hat{j} + \dfrac{\partial V}{\partial z}\hat{k} \right)

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

Calculating the partial derivatives:

∂V∂x=A(y+z)\dfrac{\partial V}{\partial x} = A(y + z)

∂V∂y=A(x+z)\dfrac{\partial V}{\partial y} = A(x + z)

∂V∂z=A(y+x)\dfrac{\partial V}{\partial z} = A(y + x)

Substituting these into the expression for the electric field:

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

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