Electrodynamics Formula Sheet & Quick Revision Guide - IIT-JEE MAINS 2027

Mastering Electrodynamics is essential for scoring high in the Physics section of IIT-JEE MAINS 2027. Electrodynamics is one of the heaviest-weighted modules in the NTA syllabus, spanning foundational vector calculus to advanced Maxwell equations and electromagnetic radiation. Below is a structured LibreTexts-style chapter breakdown containing all core formulas, differential and integral relations, vector operator identities, and exam context analysis.

---

1. Vector Calculus & Basics

Vector calculus forms the mathematical backbone of classical electrodynamics. Field calculations in Cartesian coordinates require fluency in gradient, divergence, and curl operators, alongside standard identity expansions.

Gradient, Divergence, and Curl

For a scalar field \(f(x,y,z)\) and a vector field \(\mathbf{A} = A_x\hat{x} + A_y\hat{y} + A_z\hat{z}\):

• Gradient (Cartesian): \[ abla f = \frac{\partial f}{\partial x}\hat{x} + \frac{\partial f}{\partial y}\hat{y} + \frac{\partial f}{\partial z}\hat{z}\]
• Divergence: \[ abla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\]
• Curl: \[ abla \times \mathbf{A} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}\]

Product Rules & Fundamental Vector Identities

• \( abla(fg) = f abla g + g abla f\)
• \( abla \cdot (f\mathbf{A}) = f( abla \cdot \mathbf{A}) + \mathbf{A} \cdot abla f\)
• \( abla \times (f\mathbf{A}) = f( abla \times \mathbf{A}) - \mathbf{A} \times abla f\)
• Null Identities: \( abla \times ( abla f) = 0\) (curl of gradient is identically zero)
• Null Identities: \( abla \cdot ( abla \times \mathbf{A}) = 0\) (divergence of curl is identically zero)

---

2. Electrostatics

Electrostatics deals with stationary electric charges, field intensity vectors, electrostatic potential field mappings, and partial differential equations governing potential distributions.

Coulomb's Law & Continuous Charge Distributions

The force between two point charges \(q_1\) and \(q_2\) separated by displacement vector \(\mathbf{r}\) is given by Coulomb's law:

\[\mathbf{F} = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \hat{\mathbf{r}}\]

For a continuous volume charge density \(\varrho(\mathbf{r}')\), the total electric field \(\mathbf{E}\) at position \(\mathbf{r}\) is calculated by integrating over the charge distribution volume \(d\tau'\):

\[\mathbf{E} = \frac{1}{4\pi \epsilon_0} \int \frac{\varrho(\mathbf{r}')}{r^2} \hat{\mathbf{r}} \, d\tau'\]

Gauss's Law

• Integral Form: \[\oint_{\mathcal{S}} \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\]
• Differential Form: \[ abla \cdot \mathbf{E} = \frac{\varrho}{\epsilon_0}\]

Electrostatic Potential & Differential Equations

The scalar potential \(V(\mathbf{r})\) is related to conservative electric fields via path integration and gradient operators:

\[V(\mathbf{r}) = - \int_{\mathcal{O}}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l} \quad \implies \quad \mathbf{E} = - abla V\]

Substituting \(\mathbf{E} = - abla V\) into Gauss's differential law yields Poisson's and Laplace's equations:

• Poisson's Equation: \[ abla^2 V = -\frac{\varrho}{\epsilon_0}\]
• Laplace's Equation (Charge-free region, \(\varrho = 0\)): \[ abla^2 V = 0\]

---

3. Magnetostatics

Magnetostatics governs steady electric currents, magnetic flux density vectors, vector magnetic potential fields, and circuital laws.

Lorentz Force Law & Biot-Savart Law

A particle carrying charge \(q\) moving with velocity \(\mathbf{v}\) through combined fields experiences the Lorentz force:

\[\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\]

The magnetic field \(\mathbf{B}\) generated by a steady line current \(I\) flowing along differential element \(d\mathbf{l}\) follows the Biot-Savart expression:

\[\mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}\]

Ampere's Circuital Law

• Integral Form: \[\oint_{\mathcal{P}} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\]
• Differential Form: \[ abla \times \mathbf{B} = \mu_0 \mathbf{J}\]

Magnetic Vector Potential

Because magnetic monopoles do not exist (\( abla \cdot \mathbf{B} = 0\)), the magnetic field can be expressed as the curl of a magnetic vector potential \(\mathbf{A}\):

\[\mathbf{B} = abla \times \mathbf{A}\]

Under the standard Coulomb Gauge condition, the divergence of \(\mathbf{A}\) is set to zero:

\[ abla \cdot \mathbf{A} = 0\]

---

4. Electrodynamics & Maxwell's Equations

Time-varying electromagnetic fields couple electric and magnetic phenomena, leading to Maxwell's unified equations of electrodynamics.

Faraday's Law of Induction

A time-varying magnetic field induces a non-conservative electric field described by Faraday's Law in differential form:

\[ abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\]

Ampere-Maxwell Law

Maxwell resolved the continuity equation conflict by adding displacement current density \(\mathbf{J}_d = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) to Ampere's Law:

\[ abla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\]

Summary of Maxwell's Equations (Differential Form)

1. Gauss's Law for Electricity: \( abla \cdot \mathbf{E} = \frac{\varrho}{\epsilon_0}\)
2. Gauss's Law for Magnetism: \( abla \cdot \mathbf{B} = 0\)
3. Faraday's Law of Induction: \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)
4. Ampere-Maxwell Law: \( abla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\)

---

5. Electromagnetic Waves & Radiation

In source-free charge-free vacuum space (\(\varrho = 0, \mathbf{J} = 0\)), Maxwell's equations decouple into second-order linear partial differential wave equations.

Electromagnetic Wave Equations

\[ abla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}\] \[ abla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2}\]

Speed of Light in Vacuum

The speed of electromagnetic wave propagation in vacuum space \(c\) is defined purely by fundamental free-space constants:

\[c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}\]

---

6. NTA Pattern & Chapter Weightage Analysis

Analyzing recent past year questions (PYQs) reveals key insights into how Electrodynamics is tested in IIT-JEE MAINS:

• Chapter Weightage: Electrodynamics typically carries 20% to 25% total weightage in JEE Mains Physics (approx. 5 to 7 questions out of 30).
• High-Yield Topics: Electrostatic potential field calculations, Gauss's law boundary applications, Biot-Savart integrations for finite/infinite wires, EM wave intensity, and Poynting vector parameters.
• Question Trends: Modern NTA papers heavily emphasize direct formula application, dimensional checks, and numerical value calculations involving permeability (\(\mu_0\)) and permittivity (\(\epsilon_0\)).

---

7. Frequently Asked Questions (People Also Ask)

---