Question

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

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

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.

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.

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.

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: \(V_{total} = V_A + V_B + V_C\)

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.

Key concepts

Coulomb's Law

Coulomb's Law is fundamental in understanding how electrical charges interact. It describes how two point charges exert forces upon each other. These forces can be either attractive or repulsive, depending on the sign of the charges involved. The law is given mathematically by:\[ F = \frac{k|q_1q_2|}{r^2} \]Where:- \( F \) is the force between the charges,- \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \),- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,- \( r \) is the distance between the centers of the two charges.In the case of an equilateral triangle with charges at each vertex, we are interested in the net force they exert at the center. This requires calculating forces using Coulomb’s Law for each pair of charges involved and considering how they sum up as vectors.

Electric Potential

Electric potential gives us a way to understand how charges influence each other in terms of energy rather than force. It is a scalar quantity, so it does not have direction, unlike the electric field.To find the electric potential at a point due to a point charge, the equation used is:\[ V = \frac{kq}{r} \]Where:- \( V \) is the electric potential,- \( k \) is Coulomb's constant,- \( q \) is the charge,- \( r \) is the distance from the charge to the point in question.In our triangle scenario, the potential at the center is calculated for each charge separately and then summed. Since potential is a scalar, you can directly add these values, even if the charges are negative. The result tells us how much work is done in bringing a unit positive charge from infinity to that point without acceleration.

Electric Field Vectors

Electric fields are represented by vectors that show both the magnitude and direction of the field at a point in space. To find the electric field at a point due to several charges, one must consider the direction of each contribution.Each charge creates a vector field around it. For point charges, the field due to a charge \( q \) at a distance \( r \) is directed radially and is given by:\[ E = \frac{k|q|}{r^2} \]Where:- \( E \) is the electric field,- \( k \) is Coulomb's constant,- \( q \) is the charge,- \( r \) is the distance from the charge to the point.In our triangle example, the electric fields exerted by the charges at the vertices must be considered as vectors. This involves adding them via vector addition. Direct fields from opposite charges may cancel out fully or partially, depending on their configuration, leading to a non-zero net field at the center.

Scalar and Vector Quantities

Understanding the distinction between scalar and vector quantities is critical in physics. Scalars possess only magnitude, such as mass or temperature; they do not provide any information about direction. One can simply add scalars through arithmetic addition. Vectors, however, have both magnitude and direction. Examples include force and velocity. Vectors obey different rules in mathematics, specifically vector addition, which considers both direction and magnitude to find a resultant vector. In the context of an equilateral triangle, electric potential, being a scalar, is straightforward to compute. Conversely, electric fields, requiring a vector approach, need careful consideration of the directionality and interaction of individual electric fields generated by each charge. This results in a vector sum that determines the overall field at a given point.