Introduction
Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static charges. The concepts of electrostatic potential and capacitance are fundamental to understanding various phenomena in electrostatics and are essential topics in the NEET Physics syllabus. This study note will break down these concepts into manageable sections, providing detailed explanations, examples, and tips to ensure a thorough understanding.
Electrostatic Potential
Definition and Concept
Electrostatic potential at a point in an electric field is defined as the work done in bringing a unit positive charge from infinity to that point without any acceleration.
Mathematical Expression
If $W$ is the work done in moving a charge $q$ from infinity to a point in the electric field, then the electrostatic potential $V$ at that point is given by:
$$ V = \frac{W}{q} $$
Potential Due to a Point Charge
For a point charge $Q$, the potential $V$ at a distance $r$ from the charge is given by:
$$ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} $$
where $\epsilon_0$ is the permittivity of free space.
ExampleExample Calculation: Calculate the electrostatic potential at a point 0.5 m away from a charge of $2 \times 10^{-6}$ C.
$$ V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{9 \times 10^9 , \text{N m}^2/\text{C}^2 \times 2 \times 10^{-6} , \text{C}}{0.5 , \text{m}} = 36 \times 10^3 , \text{V} $$
Potential Due to Multiple Charges
The principle of superposition applies to potentials. The total potential at a point due to multiple charges is the algebraic sum of the potentials due to individual charges.
$$ V = V_1 + V_2 + V_3 + \ldots + V_n $$
TipTo simplify calculations, remember that potential is a scalar quantity, so it can be added directly without considering direction.
Equipotential Surfaces
Definition
Equipotential surfaces are surfaces on which the potential is the same at every point. No work is required to move a charge along an equipotential surface.
Properties
- Equipotential surfaces are always perpendicular to electric field lines.
- The potential difference between any two points on an equipotential surface is zero.
The concept of equipotential surfaces is crucial in understanding the behavior of conductors in electrostatic equilibrium.
Capacitance
Definition
Capacitance is the ability of a system to store electric charge per unit potential difference. It is defined as the ratio of the charge $Q$ stored to the potential difference $V$ across the system.
$$ C = \frac{Q}{V} $$
Unit
The SI unit of capacitance is the Farad (F).
Parallel Plate Capacitor
A parallel plate capacitor consists of two parallel conductive plates separated by an insulating material (dielectric). The capacitance of a parallel plate capacitor is given by:
$$ C = \frac{\epsilon_0 \epsilon_r A}{d} $$
where:
- $\epsilon_0$ is the permittivity of free space,
- $\epsilon_r$ is the relative permittivity of the dielectric,
- $A$ is the area of one of the plates,
- $d$ is the separation between the plates.
Example Calculation: Calculate the capacitance of a parallel plate capacitor with plate area $1 , \text{m}^2$, plate separation $0.01 , \text{m}$, and air as the dielectric.
$$ C = \frac{\epsilon_0 A}{d} = \frac{8.854 \times 10^{-12} , \text{F/m} \times 1 , \text{m}^2}{0.01 , \text{m}} = 8.854 \times 10^{-10} , \text{F} $$
Energy Stored in a Capacitor
The energy $U$ stored in a capacitor is given by:
$$ U = \frac{1}{2} C V^2 $$
Derivation
The work done $dW$ to move a small charge $dq$ to the capacitor when it has a potential $V$ is:
$$ dW = V , dq $$
Since $V = \frac{q}{C}$,
$$ dW = \frac{q}{C} , dq $$
Integrating from $0$ to $Q$,
$$ W = \int_0^Q \frac{q}{C} , dq = \frac{1}{2} \frac{Q^2}{C} $$
Since $Q = CV$,
$$ W = \frac{1}{2} C V^2 $$
Common MistakeA common mistake is to forget the factor of $\frac{1}{2}$ in the energy formula, which can lead to incorrect results.
Combination of Capacitors
Capacitors in Series
The reciprocal of the equivalent capacitance $C_s$ of capacitors in series is the sum of the reciprocals of their individual capacitances:
$$ \frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots + \frac{1}{C_n} $$
Capacitors in Parallel
The equivalent capacitance $C_p$ of capacitors in parallel is the sum of their individual capacitances:
$$ C_p = C_1 + C_2 + \ldots + C_n $$
ExampleExample Calculation: Two capacitors, $C_1 = 2 , \text{F}$ and $C_2 = 3 , \text{F}$, are connected in parallel. Find the equivalent capacitance.
$$ C_p = C_1 + C_2 = 2 , \text{F} + 3 , \text{F} = 5 , \text{F} $$
Summary
Understanding electrostatic potential and capacitance is crucial for mastering electrostatics. Key points include:
- Electrostatic potential is the work done per unit charge in bringing a charge from infinity to a point.
- Equipotential surfaces are perpendicular to electric field lines and have constant potential.
- Capacitance is the ability to store charge and is defined as the charge per unit potential difference.
- Energy stored in a capacitor is given by $\frac{1}{2} C V^2$.
- Capacitors can be combined in series and parallel, with different rules for calculating the equivalent capacitance.
By mastering these concepts, you will be well-prepared for questions on electrostatics in the NEET exam.
