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Chapter 11: Problem 1573

Three charges \(2 q,-q,-q\) are located at the vertices of an equilateral triangle. At the centre of the triangle. (A) The Field is Zero but Potential is non - zero (B) The Field is non - Zero but Potential is zero (C) Both field and Potential are Zero (D) Both field and Potential are non - Zero

Short Answer

Expert verified
The electric field and potential at the center of the equilateral triangle are both non-zero. The correct statement is (D) Both field and Potential are non-Zero.

Step by step solution

01

Calculate Individual Electric Field Vectors

For calculating the electric field at the center of the equilateral triangle, we need to consider the individual electric field from each charge on the center. For an equilateral triangle, each angle = \(120^\circ\). Let's refer to vertices of the triangle as A, B, and C, with charges \(2q\), \(-q\), and \(-q\) respectively, and the center of the triangle as O. Divide the problem into three parts: 1. Electric field at O due to the charge at A (positive charge) 2. Electric field at O due to the charge at B (negative charge) 3. Electric field at O due to the charge at C (negative charge) Calculate each electric field vector separately.
02

Add Electric Field Vectors

Now, add the individual electric field vectors at point O. Since the electric field vectors due to the negative charges at vertices B and C are equal in magnitude but opposite in direction, their sum will have the same magnitude but will be directed in the opposite direction to the positive charge at vertex A. The net electric field at point O comes from the difference between the electric field due to the positive charge and the sum of the electric fields due to the negative charges.
03

Calculate Individual Electric Potentials

The next step is to calculate the electric potential at point O. Calculate the electric potentials at O due to each of the charges at A, B and C. Equation for electric potential V is given by: \(V = \frac{k*q}{r}\) where V is electric potential, k is Coulomb's constant, q is the charge, and r is the distance from the charge to the point.
04

Add Electric Potentials

Now, add the individual electric potentials at point O. Since electric potential is a scalar, we can directly add potentials due to all three charges: \(V_{total} = V_A + V_B + V_C\)
05

Determine the Correct Statement

Based on the calculations in the previous steps, we can determine which statement is correct about the electric field and potential at the center of the equilateral triangle. (A) If the Field is Zero but the Potential is non-zero; (B) If the Field is non-Zero but the Potential is zero; (C) If both the Field and Potential are Zero; (D) If both the Field and Potential are non-Zero. From our calculations, we find that the electric field at point O is non-zero, and the electric potential is also non-zero. Therefore, the correct statement is: (D) Both field and Potential are non-Zero.

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Most popular questions from this chapter

Two thin wire rings each having a radius \(R\) are placed at a distance \(d\) apart with their axes coinciding. The charges on the two rings are \(+q\) and \(-q\). The potential difference between the centers of the two rings is $\ldots .$ (A) 0 (B) $\left.\left[\mathrm{q} /\left(2 \pi \epsilon_{0}\right)\right]\left[(1 / \mathrm{R})-\left\\{1 / \sqrt{(}^{2}+\mathrm{d}^{2}\right)\right\\}\right]$ (C) $\left[\mathrm{q} /\left(4 \pi \epsilon_{0}\right)\right]\left[(1 / \mathrm{R})-\left\\{1 / \sqrt{\left. \left.\left(\mathrm{R}^{2}+\mathrm{d}^{2}\right)\right\\}\right]}\right.\right.$ (D) \(\left[(q R) /\left(4 \pi \epsilon_{0} d^{2}\right)\right]\)

The electric Potential at a point $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})\( is given by \)\mathrm{V}=-\mathrm{x}^{2} \mathrm{y}-\mathrm{x} \mathrm{z}^{3}+4\(. The electric field \)\mathrm{E}^{\boldsymbol{T}}$ at that point is \(\ldots \ldots\) (A) $i \wedge\left(2 \mathrm{xy}+\mathrm{z}^{3}\right)+\mathrm{j} \wedge \mathrm{x}^{2}+\mathrm{k} \wedge 3 \mathrm{xz}^{2}$ (B) $i \wedge 2 \mathrm{xy}+\mathrm{j} \wedge\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)+\mathrm{k} \wedge\left(3 \mathrm{xy}-\mathrm{y}^{2}\right)$ (C) \(i \wedge z^{3}+j \wedge x y z+k \wedge z^{2}\) (D) \(i \wedge\left(2 x y-z^{3}\right)+j \wedge x y^{2}+k \wedge 3 z^{2} x\)

The electric potential \(\mathrm{V}\) at any point $\mathrm{x}, \mathrm{y}, \mathrm{z}\( (all in meter) in space is given by \)\mathrm{V}=4 \mathrm{x}^{2}$ volt. The electric field at the point \((1 \mathrm{~m}, 0,2 \mathrm{~m})\) in \(\mathrm{Vm}^{-1}\) is \((\mathrm{A})+8 \mathrm{i} \wedge\) (B) \(-8 \mathrm{i} \wedge\) (C) \(-16 \mathrm{i}\) (D) \(+16 \mathrm{i}\)

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