Important Derivations for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance

Important Derivations for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance

Derivations Related to Potential due to an Electric Dipole

Derivation 1:

Electric potential due to electric dipole at a point on its axial line.

Solution:

Let ‘P’ be an axial point at distance ‘r’ from the centre of the dipole. Electric potential at point P will be $$ \begin{aligned} V & =V_1+V_2=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{(-q)}{r+a}+\frac{1}{4 \pi \varepsilon_0} \cdot \frac{q}{r-a} \\ & =\frac{q}{4 \pi \varepsilon_0}\left[\frac{1}{r-a}-\frac{1}{r+a}\right]=\frac{q}{4 \pi \varepsilon_0} \cdot \frac{2 a}{r^2-a^2} \\ & =\frac{1}{4 \pi \varepsilon_0} \cdot \frac{p}{r^2-a^2} \quad [p=q(2 a)] \end{aligned} $$ For a far away point, r>>a $$ therefore, \quad V=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{p}{r^2} \quad \text { or } \quad V \propto \frac{1}{r^2} $$ Thus, due to a dipole potential at a point is $$V \propto \frac{1}{r^2}$$.

Derivations Related to Potential Energy of System of Charges

Derivation 2:

Expression for electric potential energy of a system of three charges

To bring from infinity to , no work is required. Work done is bringing charge from infinity to is $$ q_2 V_1\left(\vec{r}_2\right)=\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r_{12}} $$ The charges and produce a potential, which at any point $p$ is given by $$ V_{12}=\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1}{r_{1 p}}+\frac{q_2}{r_{2 p}}\right) $$ Work done next in bringing from infinity to the point is $$ q_3 V_{1,2}\left(\vec{r}_3\right)=\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1 q_3}{r_{13}}+\frac{q_2 q_3}{r_{23}}\right) $$ The total work done in assembling the charges at the given location is obtained by adding the work done in steps (i) and (ii) is $$ \begin{aligned} & U=\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r_{12}}+\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1 q_3}{r_{13}}+\frac{q_2 q_3}{r_{23}}\right) \\ & =\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1 q_2}{r_{12}}+\frac{q_1 q_3}{r_{13}}+\frac{q_2 q_3}{r_{23}}\right) \end{aligned} $$

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Why students face difficulty in physics derivations?

There can be several reasons why students face difficulty in physics derivations, including:

1. Lack of foundation: A lack of understanding of the underlying concepts and principles can make it difficult to follow the logical steps in a derivation.
2. Mathematical background: Physics often involves complex mathematical calculations, and students who struggle with math may find it challenging to perform the necessary calculations.
3. Limited practice: Regular practice is key to developing the skills required for physics derivations, and students who have limited practice opportunities may struggle.
4. Poor problem-solving skills: Physics problems often require creative problem-solving skills, and students who struggle with this aspect of physics may find derivations particularly challenging.
5. Limited exposure to different problems: Physics derivations can vary widely in their level of complexity, and students who have limited exposure to different types of problems may struggle when faced with a new challenge.
6. Confusion with notation: Physics often uses a specialized notation, and students who are unfamiliar with this notation may struggle to follow the steps in a derivation.

Remembering physics derivations can be a challenge, but here are some tips that may help:

1. Practice, practice, practice: Regularly practicing physics derivations helps to build muscle memory and improve recall.
2. Visualize the process: Try to visualize the steps involved in a derivation, as well as the physical meanings behind each equation.
3. Make connections: Try to relate each step of the derivation to a concept or formula you already know, which can help to reinforce your understanding.
4. Write it down: Writing out a derivation helps to solidify your understanding and makes it easier to remember.
5. Understand the physical meaning: Try to understand the physical meaning behind each equation, which can help you to remember the derivation in a broader context.
6. Teach someone else: Teaching someone else the derivation can be a great way to reinforce your understanding and remember it more easily.
7. Use mnemonics: Creating mnemonics or acronyms can be a helpful way to remember a series of steps in a derivation.

Remember, it takes time and consistent effort to remember physics derivations. Keep practicing and seeking help when needed, and you will likely see improvement over time.

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