Electrostatic and Other IMP Chapters - MHT-CET Physics

Electrostatics, Gravitation, and Current Electricity: Key Formulas and Exam Questions

Types and Properties of Charge

Electric charge exists in two types: positive and negative. Like charges repel, unlike charges attract. The force between charges is called the electrostatic force, which can be attractive or repulsive depending on the nature of the charges.

• Unit of charge: Coulomb (C)
• Charge is quantized and conserved
• Positive–positive or negative–negative: Repel
• Positive–negative: Attract

Coulomb’s Law and Electrostatic Force

Coulomb’s Law quantifies the force between two point charges. The force is directly proportional to the product of the magnitudes…

Electrostatics, Gravitation, and Current Electricity: Key Formulas and Exam Questions

Types and Properties of Charge

Electric charge exists in two types: positive and negative. Like charges repel, unlike charges attract. The force between charges is called the electrostatic force, which can be attractive or repulsive depending on the nature of the charges.

• Unit of charge: Coulomb (C)
• Charge is quantized and conserved
• Positive–positive or negative–negative: Repel
• Positive–negative: Attract

Coulomb’s Law and Electrostatic Force

Coulomb’s Law quantifies the force between two point charges. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

• Formula: $$F = \frac{1}{4\pi\epsilon_0} \cdot k \cdot \frac{q_1 q_2}{r^2}$$
  • $q_1, q_2$: charges
  • $r$: distance between charges
  • $\epsilon_0$: permittivity of free space
  • $k$: dielectric constant (for air/vacuum, $k=1$)
• Force is a vector; direction given by position vector between charges
• In vector form: $F_{12} = \frac{1}{4\pi\epsilon_0} \cdot k \cdot \frac{q_1 q_2}{r^2} \hat{r}_{21}$

Electric Field and Electric Potential

The electric field at a point is the force per unit positive test charge at that point. Electric potential is the work done per unit charge in bringing a test charge from infinity to that point.

• Electric field: $E = \frac{F}{q_0}$
• For a point charge: $E = \frac{1}{4\pi\epsilon_0} \cdot k \cdot \frac{q}{r^2}$
• Electric potential: $V = \frac{W}{q_0}$
• For a point charge: $V = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r}$
• In potential formulas, always substitute $q$ with sign
• Relation: $E = -\frac{dV}{dx}$

Electric Dipole, Flux, and Gauss’s Law

An electric dipole consists of two equal and opposite charges separated by a distance. Electric flux measures the number of electric field lines passing through a surface. Gauss’s Law relates the total electric flux through a closed surface to the charge enclosed.

• Dipole moment: $p = q \cdot 2l$
• Electric flux: $\Phi = \vec{E} \cdot d\vec{s} = E ds \cos\theta$
• Gauss’s Law: $\oint \vec{E} \cdot d\vec{s} = \frac{q_{encl}}{\epsilon_0}$
• Flux maximum when $\theta = 0^\circ$ ($\Phi_{max} = E ds$), minimum when $\theta = 90^\circ$ ($\Phi_{min} = 0$)

Charge Distributions and Densities

Charge can be distributed linearly, over a surface, or throughout a volume. Each distribution has its own density.

• Linear charge density: $\lambda = \frac{q}{l}$ (unit: C/m)
• Surface charge density: $\sigma = \frac{q}{A}$ (unit: C/m$^2$)
• Volume charge density: $\rho = \frac{q}{V}$ (unit: C/m$^3$)
• For a sphere: $\sigma = \frac{q}{4\pi r^2}$

Dielectric Constant and Related Formulas

The dielectric constant $k$ characterizes the medium’s effect on electric interactions. Permittivity of medium $\epsilon = k \epsilon_0$.

• $k = \frac{\epsilon}{\epsilon_0}$
• $k = \frac{F_{vacuum}}{F_{medium}}$
• Force decreases as permittivity increases

Potential Energy, Work, and Torque in Electrostatics

Potential energy in an electric field and work done are closely related. Torque acts on a dipole in an electric field.

• Potential energy between two charges: $U = \frac{1}{4\pi\epsilon_0} \cdot k \cdot \frac{q_1 q_2}{r}$
• Work done: $W = Vq$
• Torque on dipole: $\tau = \vec{p} \times \vec{E}$
• Potential energy of dipole in field: $U = -\vec{p} \cdot \vec{E}$

Capacitance and Capacitors

A capacitor stores energy in the electric field between its plates. Capacitance depends on geometry and dielectric.

• Capacitance: $C = \frac{q}{V}$ (unit: Farad)
• Parallel plate capacitor: $C = \frac{k \epsilon_0 A}{d}$
• Series combination: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$
• Parallel combination: $C_{eq} = C_1 + C_2 + \dots$
• Energy stored: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{qV}{2}$
• Force between plates: $F = \frac{CV^2}{2d}$

Gravitation: Laws, Field, and Potential

Gravitational force acts between masses and is always attractive. The field and potential have direct analogies with electrostatics.

• Newton’s Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$
• $G = 6.67 \times 10^{-11}$ N m$^2$/kg$^2$ (as stated)
• Gravitational field intensity: $I = \frac{F}{m} = \frac{GM}{r^2}$
• Gravitational potential: $V = -\frac{GM}{r}$
• Potential energy: $U = -\frac{GMm}{r}$
• For small heights above surface: $g_h = g(1 - 2h/R)$ (valid up to 320 km)
• For depth $d$ below surface: $g_d = g(1 - d/R)$

Kepler’s Laws and Satellite Motion

Kepler’s laws describe planetary motion. Satellite motion involves escape and orbital velocities.

• First Law: Planets move in ellipses with the Sun at one focus.
• Second Law: Equal areas swept in equal times; $da/dt$ is constant.
• Third Law: $T^2 \propto a^3$ (where $a$ is semi-major axis)
• Escape velocity from surface: $v_{esc} = \sqrt{2gR}$
• Orbital velocity: $v_{orb} = \sqrt{gR}$
• Time period near surface: $T = 2\pi \sqrt{\frac{R}{g}}$
• Binding energy of satellite: $BE = \frac{GMm}{2(R+h)}$

Current Electricity: Laws and Circuits

Current is the rate of flow of charge. Kirchhoff’s laws govern current and voltage in circuits.

• Current: $I = \frac{dq}{dt}$
• Kirchhoff’s Current Law (KCL): Sum of currents at a junction is zero (incoming = outgoing)
• Kirchhoff’s Voltage Law (KVL): Sum of potential differences in a closed loop is zero
• Ohm’s Law: $V = IR$
• Resistance of wire: $R = \rho \frac{L}{A}$, $A = \pi r^2$
• Series resistors: $R_{eq} = R_1 + R_2 + \dots$
• Parallel resistors: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$

Wheatstone Bridge, Meter Bridge, and Potentiometer

Wheatstone bridge is used for precise resistance measurement. Meter bridge and potentiometer are practical applications.

• Wheatstone bridge balance: $\frac{P}{Q} = \frac{S}{R}$ or $PR = QS$
• Meter bridge: $\frac{X}{R} = \frac{L_x}{L_r}$, $L_x + L_r = 100$ cm
• Potentiometer principle: $V \propto L$, $E_1/E_2 = L_1/L_2$
• Internal resistance by potentiometer: $r = R \frac{L_1}{L_2 - L_1}$

Galvanometer, Ammeter, and Voltmeter Conversion

A galvanometer can be converted to an ammeter or voltmeter by connecting appropriate resistances.

• Ammeter conversion: Shunt resistance $S = \frac{I_G}{I - I_G} G$
• Voltmeter conversion: Series resistance $R = \frac{V}{I_G} - G$
• For $x\%$ of main current through galvanometer: $I_G = \frac{x}{100} I$
• Shunt for $x\%$: $S = \frac{G \cdot x}{100 - x}$

Special Cases, Shortcuts, and Exam Traps

Certain configurations and shortcuts are frequently tested. Recognize balance conditions and calculation tricks.

• In a square of four identical resistors, resistance across diagonal is $R$
• If wire is stretched $n$ times, new resistance is $n^2 R$
• In Wheatstone bridge, if balance is achieved, current through central branch is zero
• For meter bridge, balance length is independent of wire’s cross-sectional area
• Potential gradient and sensitivity of potentiometer are inversely related
• In satellite questions, escape velocity is always $\sqrt{2}$ times orbital velocity at same radius
• For gravitation, $g$ always decreases as you go above or below Earth’s surface

Exam-Focused Problem Types and Approaches

Many exam questions are direct formula applications or require ratio/percentage reasoning. Practice with previous year questions is highly recommended.

• Direct substitution in formulas for force, field, potential, capacitance, resistance
• Ratio method for changes in charge, distance, or configuration
• Recognize when to use sign conventions (especially for potential and energy)
• For Wheatstone/meter bridge, set up ratios and solve for unknowns
• For potentiometer, use proportionality of $V$ and $L$
• For satellite and gravitation, substitute given values and simplify using provided relationships