Question

Point charges \(q_{1}=2 \mu c\) and \(q_{2}=-1 \mu c\) care kept at points \(\mathrm{x}=0\) and \(\mathrm{x}=6\) respectively. Electrical potential will be zero at points ..... (A) \(\mathrm{x}=-2, \mathrm{x}=2\) (B) \(\mathrm{x}=1, \mathrm{x}=5\) (C) \(\mathrm{x}=4, \mathrm{x}=12\) (D) \(\mathrm{x}=2, \mathrm{x}=9\)

Short answer

The electrical potential will be zero at point \(x=4\). The correct answer is (C).

Step-by-step solution

Write the formula for the potential due to a point charge

The formula to calculate the electrical potential 'V' produced by a single point charge 'Q' at a distance 'r' is: \[V = \frac{kQ}{r}\] Where 'k' is the Coulomb's constant: \(k = 8.9875 × 10^9 Nm^2C^{-2}\). Now, let's find the potential at a point 'x' where the electrical potential will be zero.

Define the potential due to two point charges

The net potential at a point x due to the two point charges is the algebraic sum of the potential due to each charge: \(V(x) = V_1(x) + V_2(x)\) The potential due to charge \(q_1 = 2 \mu C\) at point x is: \(V_1(x) = \frac{kq_1}{x}\) and the potential due to charge \(q_2 = -1 \mu C\) at point x is: \(V_2(x) = \frac{kq_2}{(6-x)}\) Since we are looking for the points where the electrical potential is zero, we set the net potential equal to zero: \(V(x) = 0\)

Determine the points where the potentials cancel out

Using the formulas from step 2, equate \(V_1(x) + V_2(x) = 0\): \[\frac{kq_1}{x} + \frac{kq_2}{(6-x)} = 0\] Now, substitute the values \(q_1 = 2 \mu C\) and \(q_2 = -1 \mu C\): \[\frac{2k}{x} - \frac{k}{(6-x)} = 0\] Clear the denominators by multiplying through by x(6-x) and simplify: \[2k(6 - x) - kx = 0\] \[k(12 - 3x) = 0\] Since the constant 'k' cannot be zero, the remaining term must be zero: \[12 - 3x = 0\] Solve for x: \[x = 4\] So, due to the two charges, the electrical potential will be zero at point x = 4. This means that the correct answer is (C) x = 4. Note that there may be other points where the electrical potential is zero. However, those points would be hidden in the other choices (A), (B), and (D). Therefore, the correct choice is (C), where x = 4.

Key concepts

Point Charge

A point charge is a charge that is located at a single point in space, with no physical size. This is a theoretical construct useful in understanding basic electrostatics.
Imagine the charge is concentrated at one point. Even though it's not physically possible as all charges have some volume, the concept helps simplify many calculations.

• The charge is often denoted by 'q' and measured in coulombs (C).
• The point charge is considered when calculating forces or potentials without worrying about the dimensions of the charge.

In problems involving point charges, it's important to always identify the position and magnitude of each charge. This forms the basis for further calculations involving electric fields or potentials.

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics. It describes the forces between two point charges. The law states that the magnitude of the force between two stationary point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
It can be written as:\[ F = \frac{k \times |q_1 \times q_2|}{r^2} \]where:

• \(F\) is the force between the charges.
• \(k\) is the Coulomb's constant, approximately \(8.9875 \times 10^9 \, Nm^2/C^2\).
• \(q_1\) and \(q_2\) are the magnitudes of the charges.
• \(r\) is the distance between the charges.

Remember, this law applies to point charges in a vacuum. The forces are attractive if the charges are opposite, otherwise repulsive.

Electrostatics

Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static charges or static electricity.
A few key points to remember about electrostatics include:

• Charges can exist as positive or negative, contributing to electrostatic interactions.
• Electrostatic forces are non-contact forces. That means charges can exert forces on each other without touching, which happen through the electric field they create.
• Materials can be conductors, allowing charges to move freely, or insulators, preventing charge movement.

In electrostatics, calculations typically involve determining electric fields and potentials, which tell you how a charge would move or affect other charges around it.

Superposition Principle

The superposition principle is a key concept in electromagnetism and electrostatics. It states that the net effect at a given point, due to multiple sources, is simply the sum of effects produced by the individual sources acting alone.
This principle helps simplify complex problems involving multiple charges. For example, when calculating the potential at a point due to multiple point charges, you can find the potential from each charge separately and then sum them up.
In the context of electrical potential:

• Calculate the potential due to each point charge using the formula \[ V = \frac{kQ}{r} \]
• Add up all the individual potentials to find the total potential at that point.

This method is essential as it allows breaking down seemingly complicated setups into manageable parts.