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NEET PHYSICSELECTROSTATIC POTENTIAL AND CAPACITANCEMedium

Question

If potential in a region is expressed as V(x,y,z)=6xy−y+2yzV(x,y,z) = 6xy - y + 2yz, the electric field at point (1,1,0)(1, 1, 0) is:

A

−(3i^+5j^+3k^)-(3\hat{i} + 5\hat{j} + 3\hat{k})

B

−(6i^+5j^+2k^)-(6\hat{i} + 5\hat{j} + 2\hat{k})

C

−(2i^+3j^+k^)-(2\hat{i} + 3\hat{j} + \hat{k})

D

−(6i^+9j^+k^)-(6\hat{i} + 9\hat{j} + \hat{k})

Step-by-Step Solution

The electric field E\mathbf{E} is related to the electric potential VV by the negative gradient relationship: E=−∇V=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\mathbf{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right) [NCERT Class 12, Sec 2.6].

Given V=6xy−y+2yzV = 6xy - y + 2yz, we calculate the partial derivatives:

  1. ∂V∂x=∂∂x(6xy−y+2yz)=6y\frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(6xy - y + 2yz) = 6y
  2. ∂V∂y=∂∂y(6xy−y+2yz)=6x−1+2z\frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(6xy - y + 2yz) = 6x - 1 + 2z
  3. ∂V∂z=∂∂z(6xy−y+2yz)=2y\frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(6xy - y + 2yz) = 2y

At the point (1,1,0)(1, 1, 0), substituting x=1,y=1,z=0x=1, y=1, z=0: Ex=−(∂V∂x)=−(6(1))=−6E_x = -\left(\frac{\partial V}{\partial x}\right) = -(6(1)) = -6 Ey=−(∂V∂y)=−(6(1)−1+2(0))=−(5)=−5E_y = -\left(\frac{\partial V}{\partial y}\right) = -(6(1) - 1 + 2(0)) = -(5) = -5

  • Ez=−(∂V∂z)=−(2(1))=−2E_z = -\left(\frac{\partial V}{\partial z}\right) = -(2(1)) = -2

Thus, E=−6i^−5j^−2k^=−(6i^+5j^+2k^)\mathbf{E} = -6\hat{i} - 5\hat{j} - 2\hat{k} = -(6\hat{i} + 5\hat{j} + 2\hat{k}) N/C.

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from ELECTROSTATIC POTENTIAL AND CAPACITANCE. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSELECTROSTATIC POTENTIAL AND CAPACITANCEpotentialregionexpressedelectric

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