This Electrostatics Formula Sheet provides a concise yet comprehensive summary of key concepts and formulas essential for JEE Main preparation. Mastering these principles will enable you to tackle a wide range of problems involving charges, electric fields, potential, and capacitance.
Key Concepts
Electric Charge: An intrinsic property of matter that gives rise to electric forces. It is quantized ($q = \pm ne$) and conserved. Like charges repel, unlike charges attract. Coulomb's Law: Describes the force between two point charges. It acts along the line joining the charges. Electric Field: The region around a charged object where another charged object experiences an electric force. It can be visualized using electric field lines. Electric Potential: The work done per unit positive test charge to bring it from infinity to a point in an electric field without acceleration. It is a scalar quantity. Electric Potential Energy: The work done to assemble a system of charges from infinity to their current configuration. Gauss's Law: A fundamental law relating the electric flux through a closed surface to the net charge enclosed within that surface. It is particularly useful for finding electric fields of symmetric charge distributions. Conductors: Materials with a large number of free charge carriers (electrons) that can move easily. In electrostatic equilibrium, the electric field inside a conductor is zero, and charges reside on its surface. Insulators (Dielectrics): Materials with very few free charge carriers. When placed in an electric field, they become polarized. Capacitance: The ability of a conductor or a system of conductors (capacitor) to store electric charge. It is defined as the ratio of the charge stored to the potential difference across it. Dielectric Strength: The maximum electric field a dielectric material can withstand without breaking down (losing its insulating properties).
Formulas & Results
1. Coulomb's Law: The force between two point charges $q_1$ and $q_2$ separated by a distance $r$ is: $$F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}$$ where $k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2$ is Coulomb's constant, and $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2$ is the permittivity of free space.
2. Electric Field:
- Due to a point charge $Q$ at a distance $r$: $$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$
- Electric field due to an electric dipole of dipole moment $\vec{p} = q(2a)\hat{p}$:
- On its axial line (at distance $r$ from center, $r \gg a$): $$ \vec{E}_{\text{axial}} = \frac{1}{4\pi\epsilon_0} \frac{2\vec{p}}{r^3} $$
- On its equatorial line (at distance $r$ from center, $r \gg a$): $$ \vec{E}_{\text{equatorial}} = -\frac{1}{4\pi\epsilon_0} \frac{\vec{p}}{r^3} $$
- Due to an infinitely long thin straight wire with linear charge density $\lambda$: $$E = \frac{\lambda}{2\pi\epsilon_0 r}$$
- Due to an infinite plane sheet with surface charge density $\sigma$: $$E = \frac{\sigma}{2\epsilon_0}$$
- Due to a uniformly charged thin spherical shell (radius $R$, charge $Q$):
- Outside ($r > R$): $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$
- On surface ($r = R$): $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^2}$
- Inside ($r < R$): $E = 0$
- Due to a uniformly charged solid sphere (radius $R$, charge $Q$):
- Outside ($r > R$): $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$
- On surface ($r = R$): $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^2}$
- Inside ($r < R$): $E = \frac{1}{4\pi\epsilon_0} \frac{Qr}{R^3}$
3. Electric Potential:
- Relation between electric field and potential: $$ \vec{E} = -\nabla V \quad \text{ or } \quad E = -\frac{dV}{dr} $$
- Due to a point charge $Q$ at a distance $r$: $$V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$$
- Due to an electric dipole (axial point, $r \gg a$): $$V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$$ where $\theta$ is the angle between $\vec{p}$ and $\vec{r}$.
- Electric potential energy of two point charges $q_1, q_2$ separated by $r$: $$U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$$
- Potential energy of a dipole in a uniform electric field: $$U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$$
4. Gauss's Law: $$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$
5. Capacitance:
- Definition: $C = \frac{Q}{V}$
- Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ (in vacuum)
- Parallel Plate Capacitor with dielectric (dielectric constant $K$): $C = \frac{K\epsilon_0 A}{d}$
- Spherical Capacitor (radii $R_1, R_2$): $C = 4\pi\epsilon_0 \frac{R_1 R_2}{R_2 - R_1}$
- Isolated Spherical Conductor (radius $R$): $C = 4\pi\epsilon_0 R$
- Cylindrical Capacitor (length $L$, radii $R_1, R_2$): $C = \frac{2\pi\epsilon_0 L}{\ln(R_2/R_1)}$
6. Combination of Capacitors:
- Series: $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ ($Q$ is same across each, $V = V_1 + V_2 + \dots$)
- Parallel: $C_{\text{eq}} = C_1 + C_2 + \dots$ ($V$ is same across each, $Q = Q_1 + Q_2 + \dots$)
7. Energy Stored in a Capacitor: $$U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$$
- Energy density (energy per unit volume) in a uniform electric field: $$u_E = \frac{1}{2}\epsilon_0 E^2$$
8. Force on Dielectric in Capacitor: When a dielectric slab is gradually inserted into a parallel plate capacitor, the force attracting the slab is: $$F = \frac{Q^2}{2C^2} \frac{dC}{dx} = \frac{1}{2}V^2 \frac{dC}{dx}$$ For a rectangular slab of width $b$ inserted a distance $x$ into a parallel plate capacitor with plate dimensions $L \times b$: $$F = \frac{\epsilon_0 b(K-1)}{2d} V^2$$ (if battery connected) $$F = \frac{Q^2 d (K-1)}{2 \epsilon_0 A [d + x(K-1)]^2} x_i$$ (if battery disconnected) - this is complicated for JEE Advanced, simpler for constant voltage.
Worked Examples
1. Example: Electric Field of a Dipole An electric dipole has a dipole moment of $4 \times 10^{-9} \text{ C m}$ directed along the positive x-axis. Calculate the electric field at a point (1 m, 0) on the axial line.
Solution: The point (1 m, 0) is on the axial line of the dipole. The dipole moment $\vec{p}$ is along the positive x-axis. The distance $r = 1 \text{ m}$. For an axial point, the electric field is given by: $$ \vec{E}_{\text{axial}} = \frac{1}{4\pi\epsilon_0} \frac{2\vec{p}}{r^3} $$ Given $p = 4 \times 10^{-9} \text{ C m}$ and $r = 1 \text{ m}$. We know $\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2$. $$ E = (9 \times 10^9) \frac{2 \times (4 \times 10^{-9})}{(1)^3} $$ $$ E = 9 \times 10^9 \times 8 \times 10^{-9} $$ $$ E = 72 \text{ N/C} $$ Since $\vec{p}$ is along the positive x-axis, the electric field $\vec{E}$ at an axial point along the positive x-axis will also be along the positive x-axis. So, $\vec{E} = 72 \hat{i} \text{ N/C}$.
2. Example: Capacitance and Energy Storage A parallel plate capacitor has plates of area $100 \text{ cm}^2$ and separation $1 \text{ mm}$. A potential difference of $100 \text{ V}$ is applied across it. (a) Calculate the capacitance of the capacitor. (b) Calculate the energy stored in the capacitor.
Solution: Given: Area $A = 100 \text{ cm}^2 = 100 \times 10^{-4} \text{ m}^2 = 10^{-2} \text{ m}^2$. Separation $d = 1 \text{ mm} = 1 \times 10^{-3} \text{ m}$. Potential difference $V = 100 \text{ V}$. Permittivity of free space $\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$.
(a) Capacitance $C$: $$ C = \frac{\epsilon_0 A}{d} $$ $$ C = \frac{(8.85 \times 10^{-12} \text{ F/m}) \times (10^{-2} \text{ m}^2)}{(1 \times 10^{-3} \text{ m})} $$ $$ C = 8.85 \times 10^{-11} \text{ F} $$ $$ C = 88.5 \text{ pF} $$
(b) Energy stored $U$: $$ U = \frac{1}{2}CV^2 $$ $$ U = \frac{1}{2} (8.85 \times 10^{-11} \text{ F}) (100 \text{ V})^2 $$ $$ U = \frac{1}{2} (8.85 \times 10^{-11}) (10^4) $$ $$ U = 4.425 \times 10^{-7} \text{ J} $$
3. Example: Gauss's Law Application A solid non-conducting sphere of radius $R$ has a total charge $Q$ uniformly distributed throughout its volume. Find the electric field at a point inside the sphere at a distance $r < R$ from the center.
Solution: Consider a spherical Gaussian surface of radius $r$ ($r < R$) concentric with the solid sphere. According to Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$. Due to spherical symmetry, $\vec{E}$ is radial and has the same magnitude at all points on the Gaussian surface. Also, $\vec{E}$ is parallel to $d\vec{A}$. So, $\oint E dA = E \cdot 4\pi r^2$.
Now, we need to find $Q_{\text{enclosed}}$. The charge is uniformly distributed throughout the volume of the sphere. Total volume of the sphere $= \frac{4}{3}\pi R^3$. Volume charge density $\rho = \frac{Q}{\frac{4}{3}\pi R^3}$. The charge enclosed within the Gaussian surface of radius $r$ is: $$ Q_{\text{enclosed}} = \rho \times (\text{volume of Gaussian sphere}) = \frac{Q}{\frac{4}{3}\pi R^3} \times \frac{4}{3}\pi r^3 $$ $$ Q_{\text{enclosed}} = Q \frac{r^3}{R^3} $$ Applying Gauss's Law: $$ E \cdot 4\pi r^2 = \frac{1}{\epsilon_0} \left( Q \frac{r^3}{R^3} \right) $$ $$ E = \frac{1}{4\pi\epsilon_0} \frac{Q r^3}{R^3 r^2} $$ $$ E = \frac{1}{4\pi\epsilon_0} \frac{Qr}{R^3} $$
Common Mistakes
- Vector vs. Scalar: Confusing electric field (vector) with electric potential (scalar) calculations. For instance, directly adding potentials when superposition principle requires vector addition for fields.
- Sign Errors: Incorrectly handling signs of charges in potential energy calculations or for magnitude and direction of force/field. Remember convention: positive potential for positive charge, negative for negative.
- Units: Not converting units to SI (e.g., cm to m, $\mu$C to C) before calculation.
- Gauss's Law: Applying Gauss's Law incorrectly to non-symmetric charge distributions or choosing inappropriate Gaussian surfaces.
- Capacitor Combinations: Mixing up formulas for series and parallel combinations of capacitors (e.g., using reciprocal summation for parallel).
- Dielectrics: Forgetting that placing a dielectric in a capacitor (connected to a battery) increases charge but keeps voltage constant, or (disconnected battery) keeps charge constant but reduces voltage.
- Internal Points in Conductors: Failing to remember that the electric field inside a conductor in electrostatic equilibrium is always zero, and potential is constant throughout.
- Infinite vs. Point Charge: Using point charge field formula for infinite charge distributions or vice-versa.
Quick Revision Checklist
- Coulomb's Law: Force between point charges.
- Electric Field: Definition, formulas for point charge, dipole (axial/equatorial), infinite line, infinite plane, spherical shell, solid sphere.
- Electric Potential: Definition, relation with E-field, formulas for point charge, dipole, potential energy of charge systems and dipoles.
- Gauss's Law: Statement, $\Phi_E = Q_{enclosed}/\epsilon_0$, and its applications for symmetric geometries.
- Conductors: Properties in electrostatic equilibrium (E=0 inside, charge on surface, constant potential).
- Capacitance: Definition, formulas for parallel plate, spherical, cylindrical capacitors. Effect of dielectric.
- Capacitor Combinations: Series and parallel rules for equivalent capacitance.
- Energy Storage: Formulas for energy in a capacitor ($U = \frac{1}{2}CV^2$) and energy density.
- Dipole in E-field: Torque ($\vec{\tau} = \vec{p} \times \vec{E}$) and potential energy ($U = -\vec{p} \cdot \vec{E}$).
