Electrostatic Potential: Numerical Problems

1. Potential from a Symmetric System of Charges

Calculating potential is often simplified when dealing with symmetric charge distributions. Because potential is a scalar quantity, we don't need to worry about vector components, which makes the algebra much more straightforward compared to calculating electric fields.

A classic example is finding the potential at the center of a regular polygon with identical charges at its vertices. Let's look at the case of a regular hexagon.

The logic here is simple:

• All charges are identical ($+q$).
• The distance from the center O to each vertex is the same ($d$).
• Therefore, the potential at the center due to each charge is identical: $V_{single}=\frac{1}{4\pi {ϵ}_0}\frac{q}{d}$.
• Using the superposition principle, the total potential is just six times this value: $V_{total}=6\times V_{single}$.

To see a step-by-step numerical solution for this exact type of problem (NCERT Exercise 2.2), watch the following video.

2. Finding Points of Zero Potential

A very common question in exams asks you to find the point (or points) on the line joining two charges where the net electric potential is zero. This happens when the positive potential from one charge is exactly cancelled by the negative potential from the other. This requires one charge to be positive and the other negative.

Let's consider two charges, $q_A$ and $q_B$, separated by a distance $d$. We want to find a point P where the total potential $V_P=V_A+V_B=0$. This means:

where $r_A$ and $r_B$ are the distances of point P from $q_A$ and $q_B$, respectively.

There are generally two possible locations for the zero potential point:

1. Between the charges: This point will be closer to the charge with the smaller magnitude.
2. Outside the charges: On the line joining them, this point will be on the side of the charge with the smaller magnitude.

The following video provides an excellent, detailed walkthrough of solving such a problem (NCERT Exercise 2.1), including the reasoning for checking both possible locations.

Here is a static image that illustrates the calculation for a similar problem.

3. Work Done and Potential Difference

The physical significance of potential lies in its connection to work. The work done by an external force to move a charge $q$ from a point A to a point B is given by:

This work is equal to the change in the potential energy of the charge, $\Delta U$. Conversely, the work done by the electric field is the negative of this:

Let's apply this to a problem from a previous year's exam.

4. Guided Practice

Now it's your turn to practice. Your ability to solve these problems relies on a solid understanding of the formulas and a systematic approach. The following resource from CK-12 provides several worked-out examples that you can use to test your understanding.

For additional practice, you can also refer to the solved examples in your NCERT textbook. The methods shown in the videos above align perfectly with the textbook's approach.

Conclusion

Today we've shifted our focus from deriving formulas to applying them, which is essential for exam success. You've seen how the superposition principle is used to solve problems involving multiple charges and how potential difference dictates the work done in moving charges.

Key takeaways from our problem-solving session:

• Symmetry is your friend: For symmetric charge distributions, the calculation of total potential can be greatly simplified.
• Scalar Addition: Remember that potential is a scalar. You add the values algebraically, including their signs, without worrying about directions.
• Finding Zero Potential: When finding a point of zero potential between two opposite charges, always check for solutions both between the charges and outside the charge with the smaller magnitude.
• Work and Energy: The work done is directly proportional to the change in potential ($W=q\Delta V$). The work done by the field is equal to the decrease in potential energy ($W_{field}=-\Delta U$).

In our next lesson, we will introduce the concept of equipotential surfaces. These are surfaces of constant potential, and understanding them provides a powerful way to visualize electric fields. We've already seen an example: the equatorial plane of a dipole is an equipotential surface where V=0. We will explore this idea in more detail.