Electrodynamics Notes FE Sem 1 Applied Physics | MU

Every electrical machine, antenna and communication link is described by four equations written by James Clerk Maxwell. This module first builds the mathematical tools needed to read them: the gradient of a scalar field and the divergence and curl of a vector field in Cartesian coordinates. It then states Gauss's law, Ampere's circuital law and Faraday's law, links surface, volume and line integrals through the divergence and Stokes theorems, and puts everything together as Maxwell's equations in integral and point form with their physical significance. It maps to course outcome CO4, and both short numericals on grad, div, curl and descriptive questions on Maxwell's equations are common.

Summary

Gradient: The del operator in Cartesian coordinates is \(\nabla = \hat{i}\,\partial/\partial x + \hat{j}\,\partial/\partial y + \hat{k}\,\partial/\partial z\). Acting on a scalar field \(\phi(x,y,z)\) it gives the gradient \(\nabla\phi = \hat{i}\,\partial\phi/\partial x + \hat{j}\,\partial\phi/\partial y + \hat{k}\,\partial\phi/\partial z\), a vector. Its direction is the direction in which \(\phi\) increases fastest and its magnitude is that maximum rate of increase; it is normal to the level surface \(\phi\) = constant. Example: the electric field is the negative gradient of potential, \(\vec{E} = -\nabla V\). The rate of change of \(\phi\) in the direction of a unit vector \(\hat{a}\) (directional derivative) is \(\nabla\phi\cdot\hat{a}\).

Divergence: For a vector field \(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\), the divergence \(\nabla\cdot\vec{A} = \partial A_x/\partial x + \partial A_y/\partial y + \partial A_z/\partial z\) is a scalar. Physically it is the net outward flux of \(\vec{A}\) per unit volume at a point: positive divergence means a source (field lines diverge out), negative means a sink. A field with \(\nabla\cdot\vec{A} = 0\) everywhere is called solenoidal; the magnetic field is always solenoidal, which expresses the absence of magnetic monopoles.

Curl: The curl of \(\vec{A}\) is the vector \(\nabla\times\vec{A}\) given by the determinant with rows \(\hat{i}, \hat{j}, \hat{k}\); \(\partial/\partial x, \partial/\partial y, \partial/\partial z\); \(A_x, A_y, A_z\), that is \(\hat{i}(\partial A_z/\partial y - \partial A_y/\partial z) + \hat{j}(\partial A_x/\partial z - \partial A_z/\partial x) + \hat{k}(\partial A_y/\partial x - \partial A_x/\partial y)\). Physically it measures the rotation or circulation of the field per unit area about a point; its direction is the axis of rotation by the right hand rule. A field with zero curl is irrotational (conservative), for example the electrostatic field, which can therefore be written as the gradient of a potential.

Gauss's law: The total electric flux through any closed surface equals \(1/\varepsilon_0\) times the total charge enclosed: \(\oint \vec{E}\cdot d\vec{S} = Q_{enc}/\varepsilon_0\), or in terms of the displacement vector \(\vec{D} = \varepsilon\vec{E}\), \(\oint \vec{D}\cdot d\vec{S} = Q_{enc}\). It holds for any shape of surface and is most useful for symmetric charge distributions, for example it gives the field of a point charge \(E = q/4\pi\varepsilon_0 r^2\). For magnetism the corresponding law is \(\oint \vec{B}\cdot d\vec{S} = 0\), since there are no isolated magnetic poles.

Ampere's circuital law: The line integral of the magnetic field around any closed path equals \(\mu_0\) times the total current threading the path: \(\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}\), or \(\oint \vec{H}\cdot d\vec{l} = I_{enc}\) with \(\vec{B} = \mu\vec{H}\). For a long straight wire it gives \(B = \mu_0 I/2\pi r\). Maxwell noticed that this law, as written, contradicts the conservation of charge for time varying fields (for example between capacitor plates, where conduction current is interrupted), and added the displacement current \(\partial\vec{D}/\partial t\) term so that \(\oint \vec{H}\cdot d\vec{l} = I + \int (\partial\vec{D}/\partial t)\cdot d\vec{S}\).

Faraday's law: Whenever the magnetic flux \(\Phi_B = \int \vec{B}\cdot d\vec{S}\) linked with a circuit changes, an emf is induced equal to the negative rate of change of flux: \(e = -d\Phi_B/dt\), and for N turns \(e = -N\,d\Phi_B/dt\). The minus sign is Lenz's law: the induced current opposes the change producing it. Since the emf is the line integral of the induced electric field, \(\oint \vec{E}\cdot d\vec{l} = -(d/dt)\int \vec{B}\cdot d\vec{S}\); a changing magnetic field produces a circulating (non-conservative) electric field. Generators, transformers and induction motors work on this law.

Divergence theorem and Stokes theorem: The Gauss divergence theorem converts a closed surface integral into a volume integral: \(\oint_S \vec{A}\cdot d\vec{S} = \int_V (\nabla\cdot\vec{A})\,dV\); the total outward flux through a closed surface equals the total divergence inside. Stokes theorem converts a closed line integral into a surface integral: \(\oint_C \vec{A}\cdot d\vec{l} = \int_S (\nabla\times\vec{A})\cdot d\vec{S}\), where S is any surface bounded by C; the circulation round a loop equals the total curl through the surface. These theorems let us turn the integral laws of Gauss, Ampere and Faraday into point (differential) form.

Maxwell's equations and their significance: In integral form: (1) \(\oint \vec{D}\cdot d\vec{S} = \int_V \rho\,dV\); (2) \(\oint \vec{B}\cdot d\vec{S} = 0\); (3) \(\oint \vec{E}\cdot d\vec{l} = -\int_S (\partial\vec{B}/\partial t)\cdot d\vec{S}\); (4) \(\oint \vec{H}\cdot d\vec{l} = \int_S (\vec{J} + \partial\vec{D}/\partial t)\cdot d\vec{S}\). In point form: \(\nabla\cdot\vec{D} = \rho\), \(\nabla\cdot\vec{B} = 0\), \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\), \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\). Significance: the first says charges are sources of electric field; the second says magnetic field lines are closed and monopoles do not exist; the third says a time varying magnetic field produces an electric field; the fourth says magnetic fields are produced by conduction current and by a changing electric field (displacement current). Together they predict electromagnetic waves travelling at \(c = 1/\sqrt{\mu_0\varepsilon_0}\), showing that light is an electromagnetic wave, and they underpin all of electrical engineering and communication.

Mind map

Converting the integral laws into Maxwell's point form

Important definitions

Formulas

Key points to remember

• Gradient acts on a scalar and gives a vector; divergence acts on a vector and gives a scalar; curl acts on a vector and gives a vector.
• Gradient is perpendicular to the level surfaces of the scalar field.
• Div = 0: solenoidal; curl = 0: irrotational.
• Curl of a gradient is always zero, and divergence of a curl is always zero.
• Gauss's law relates flux through a closed surface to enclosed charge.
• Ampere's law relates circulation of H to enclosed current; Maxwell added displacement current.
• Faraday's law: changing magnetic flux induces an emf; the minus sign is Lenz's law.
• Divergence theorem links surface and volume integrals; Stokes theorem links line and surface integrals.
• Maxwell's equations predict electromagnetic waves travelling at c = 1/√(μ0ε0) = 3 x 10^8 m/s.
• Point form holds at a point; integral form holds over a region, surface or loop.

Solved examples

Example 1. Find the gradient of \(\phi = 3x^2y - y^3z^2\) at the point (1, −2, −1) and its magnitude.

\(\partial\phi/\partial x = 6xy\), \(\partial\phi/\partial y = 3x^2 - 3y^2z^2\), \(\partial\phi/\partial z = -2y^3z\). At (1, −2, −1): \(6(1)(-2) = -12\); \(3(1) - 3(4)(1) = -9\); \(-2(-8)(-1) = -16\). So \(\nabla\phi = -12\hat{i} - 9\hat{j} - 16\hat{k}\). Magnitude \(= \sqrt{144 + 81 + 256} = \sqrt{481} = 21.93\); this is the maximum rate of change of φ at that point.

Example 2. Find the divergence of \(\vec{A} = x^2y\,\hat{i} + yz^2\,\hat{j} - 2xz\,\hat{k}\) at (1, 1, 1).

\(\nabla\cdot\vec{A} = \partial(x^2y)/\partial x + \partial(yz^2)/\partial y + \partial(-2xz)/\partial z = 2xy + z^2 - 2x\). At (1, 1, 1): \(2 + 1 - 2 = 1\). The point acts as a weak source.

Example 3. Find the curl of \(\vec{A} = xy\,\hat{i} + yz\,\hat{j} + zx\,\hat{k}\).

\(\hat{i}\) component: \(\partial(zx)/\partial y - \partial(yz)/\partial z = 0 - y = -y\). \(\hat{j}\) component: \(\partial(xy)/\partial z - \partial(zx)/\partial x = 0 - z = -z\). \(\hat{k}\) component: \(\partial(yz)/\partial x - \partial(xy)/\partial y = 0 - x = -x\). So \(\nabla\times\vec{A} = -y\hat{i} - z\hat{j} - x\hat{k}\).

Example 4. Show that \(\vec{F} = (y+z)\hat{i} + (z+x)\hat{j} + (x+y)\hat{k}\) is both irrotational and solenoidal.

Curl: \(\hat{i}(\partial(x+y)/\partial y - \partial(z+x)/\partial z) = \hat{i}(1-1) = 0\); \(\hat{j}(\partial(y+z)/\partial z - \partial(x+y)/\partial x) = \hat{j}(1-1) = 0\); \(\hat{k}(\partial(z+x)/\partial x - \partial(y+z)/\partial y) = \hat{k}(1-1) = 0\). So \(\nabla\times\vec{F} = 0\): irrotational. Divergence: \(\partial(y+z)/\partial x + \partial(z+x)/\partial y + \partial(x+y)/\partial z = 0\): solenoidal.

Example 5. Find the constant a so that \(\vec{A} = (x + 3y)\hat{i} + (y - 2z)\hat{j} + (x + az)\hat{k}\) is solenoidal.

\(\nabla\cdot\vec{A} = 1 + 1 + a = 0\), so \(a = -2\).

Example 6. Verify the divergence theorem for \(\vec{A} = x\hat{i} + y\hat{j} + z\hat{k}\) over the unit cube \(0 \le x, y, z \le 1\).

Volume side: \(\nabla\cdot\vec{A} = 3\), so \(\int_V 3\,dV = 3\times1 = 3\). Surface side: on face x = 1 the outward normal is \(\hat{i}\) and \(\vec{A}\cdot\hat{i} = x = 1\), flux = 1; on face x = 0, \(\vec{A}\cdot(-\hat{i}) = -x = 0\), flux 0. Similarly the y = 1 and z = 1 faces give 1 each and the others zero. Total flux = 3. Both sides equal 3, so the theorem is verified.

Example 7. Charges of 2 μC, −5 μC and 8 μC lie inside a closed surface. Find the total electric flux through it.

By Gauss's law \(\Phi_E = Q_{enc}/\varepsilon_0 = (2 - 5 + 8)\times10^{-6}/8.854\times10^{-12} = 5\times10^{-6}/8.854\times10^{-12} = 5.65\times10^{5}\) N m²/C. Charges outside would contribute nothing.

Example 8. The magnetic flux through each turn of a 100 turn coil is \(\Phi = (5t^2 + 3t)\) mWb. Find the induced emf at t = 2 s.

\(d\Phi/dt = (10t + 3)\) mWb/s = 23 mWb/s at t = 2 s. \(e = -N\,d\Phi/dt = -100\times23\times10^{-3} = -2.3\) V. Magnitude 2.3 V; the sign shows it opposes the increase in flux.

Important questions and answers

Q1. Explain the physical significance of gradient, divergence and curl. (6 marks)

Gradient: for a scalar field φ, \(\nabla\phi\) is a vector pointing in the direction of the greatest rate of increase of φ, with magnitude equal to that rate; it is normal to surfaces of constant φ. Example: \(\vec{E} = -\nabla V\), field points from high to low potential. Divergence: for a vector field \(\vec{A}\), \(\nabla\cdot\vec{A}\) is the net outward flux per unit volume around a point. Positive: source (lines spread out), negative: sink, zero: solenoidal (as many lines enter as leave), e.g. \(\nabla\cdot\vec{B} = 0\). Curl: \(\nabla\times\vec{A}\) measures the circulation (tendency to rotate) per unit area; direction is the axis of rotation by the right hand rule. Zero curl: irrotational or conservative field (electrostatic field). Example: velocity field of a rotating wheel has non-zero curl; magnetic field around a current has \(\nabla\times\vec{H} = \vec{J}\).

Q2. State and explain Gauss's law in electrostatics. Use it to find the electric field of a point charge. (5 marks)

Statement: the total electric flux through any closed surface is \(1/\varepsilon_0\) times the net charge enclosed: \(\oint \vec{E}\cdot d\vec{S} = Q/\varepsilon_0\) (or \(\oint \vec{D}\cdot d\vec{S} = Q\)). The surface (Gaussian surface) can be of any shape; charges outside give zero net flux. Point charge: choose a sphere of radius r centred on q. By symmetry E is radial and of equal magnitude everywhere on the sphere, parallel to dS. So \(\oint E\,dS = E(4\pi r^2) = q/\varepsilon_0\), hence \(E = q/(4\pi\varepsilon_0 r^2)\), which is Coulomb's law. Point form using the divergence theorem: \(\nabla\cdot\vec{D} = \rho\).

Q3. State Ampere's circuital law. Why did Maxwell modify it? (5 marks)

Statement: the line integral of the magnetic field intensity around any closed path equals the total current enclosed: \(\oint \vec{H}\cdot d\vec{l} = I\), or \(\oint \vec{B}\cdot d\vec{l} = \mu_0 I\). By Stokes theorem, point form \(\nabla\times\vec{H} = \vec{J}\). Problem: taking divergence, \(\nabla\cdot(\nabla\times\vec{H}) = 0\) gives \(\nabla\cdot\vec{J} = 0\), but the continuity equation requires \(\nabla\cdot\vec{J} = -\partial\rho/\partial t\), which is non-zero for time varying charge (e.g. a charging capacitor, where no conduction current flows between the plates yet a magnetic field exists). Maxwell's fix: using \(\rho = \nabla\cdot\vec{D}\), \(\nabla\cdot(\vec{J} + \partial\vec{D}/\partial t) = 0\), so he added the displacement current density \(\partial\vec{D}/\partial t\): \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\). This means a changing electric field also produces a magnetic field, which made electromagnetic waves possible.

Q4. State Faraday's law of electromagnetic induction and derive its point form. (4 marks)

Statement: the emf induced in a closed circuit equals the negative rate of change of magnetic flux linked with it: \(e = -d\Phi_B/dt\); the minus sign (Lenz's law) shows the induced current opposes the change. The emf is \(\oint \vec{E}\cdot d\vec{l}\) and \(\Phi_B = \int \vec{B}\cdot d\vec{S}\), so \(\oint \vec{E}\cdot d\vec{l} = -\int (\partial\vec{B}/\partial t)\cdot d\vec{S}\) for a stationary loop. By Stokes theorem \(\oint \vec{E}\cdot d\vec{l} = \int (\nabla\times\vec{E})\cdot d\vec{S}\). Comparing for an arbitrary surface, \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\).

Q5. State the Gauss divergence theorem and Stokes theorem. (3 marks)

Divergence theorem: the surface integral of the normal component of a vector field over a closed surface equals the volume integral of its divergence over the enclosed volume, \(\oint_S \vec{A}\cdot d\vec{S} = \int_V \nabla\cdot\vec{A}\,dV\). Stokes theorem: the line integral of a vector field around a closed curve equals the surface integral of its curl over any surface bounded by that curve, \(\oint_C \vec{A}\cdot d\vec{l} = \int_S (\nabla\times\vec{A})\cdot d\vec{S}\).

Q6. Write Maxwell's equations in integral and point form and explain the significance of each. (6 marks)

(1) Gauss's law for electricity: \(\oint \vec{D}\cdot d\vec{S} = \int \rho\,dV\); \(\nabla\cdot\vec{D} = \rho\). Electric field lines start and end on charges; charge is the source of electric flux. (2) Gauss's law for magnetism: \(\oint \vec{B}\cdot d\vec{S} = 0\); \(\nabla\cdot\vec{B} = 0\). Net magnetic flux through any closed surface is zero; magnetic monopoles do not exist and B lines form closed loops. (3) Faraday's law: \(\oint \vec{E}\cdot d\vec{l} = -\int (\partial\vec{B}/\partial t)\cdot d\vec{S}\); \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\). A time varying magnetic field produces a circulating electric field; basis of generators and transformers. (4) Ampere-Maxwell law: \(\oint \vec{H}\cdot d\vec{l} = \int (\vec{J} + \partial\vec{D}/\partial t)\cdot d\vec{S}\); \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\). Magnetic field is produced by conduction current and by a changing electric field. Overall: they unify electricity, magnetism and optics, and predict electromagnetic waves moving at \(c = 1/\sqrt{\mu_0\varepsilon_0}\).

Q7. Derive Maxwell's first equation in point form from Gauss's law. (3 marks)

Gauss's law: \(\oint_S \vec{D}\cdot d\vec{S} = Q = \int_V \rho\,dV\). By the divergence theorem the left side equals \(\int_V \nabla\cdot\vec{D}\,dV\). So \(\int_V (\nabla\cdot\vec{D} - \rho)\,dV = 0\) for every volume, which is possible only if the integrand is zero: \(\nabla\cdot\vec{D} = \rho\).

Q8. What is meant by solenoidal and irrotational fields? Give one example of each. (3 marks)

A vector field \(\vec{A}\) is solenoidal if \(\nabla\cdot\vec{A} = 0\) everywhere: it has no sources or sinks and its lines are closed; example: magnetic field \(\vec{B}\). It is irrotational if \(\nabla\times\vec{A} = 0\) everywhere: it has no circulation, its line integral round any closed path is zero and it can be written as the gradient of a scalar; example: electrostatic field \(\vec{E} = -\nabla V\).

Q9. Show that the field \(\vec{E} = yz\hat{i} + zx\hat{j} + xy\hat{k}\) is conservative. (3 marks)

\(\hat{i}\): \(\partial(xy)/\partial y - \partial(zx)/\partial z = x - x = 0\). \(\hat{j}\): \(\partial(yz)/\partial z - \partial(xy)/\partial x = y - y = 0\). \(\hat{k}\): \(\partial(zx)/\partial x - \partial(yz)/\partial y = z - z = 0\). Curl is zero, so the field is irrotational, i.e. conservative; indeed \(\vec{E} = \nabla(xyz) = -\nabla V\) with potential \(V = -xyz\) (plus a constant).

Q10. If \(\vec{D} = 3x^2\hat{i} + 2y\hat{j} + z\hat{k}\) C/m², find the volume charge density at (1, 1, 1). (3 marks)

From Maxwell's first equation \(\rho = \nabla\cdot\vec{D} = 6x + 2 + 1\). At (1, 1, 1): \(\rho = 9\) C/m³.

Exam tips

• Write the curl as a 3 x 3 determinant first; it avoids sign errors in the j component.
• Always state what each operator acts on and what it returns (scalar or vector); examiners deduct for dotting a scalar.
• For Maxwell's equations write both forms side by side in a table with names and significance; this layout gets full marks quickly.
• Mention displacement current and the charging capacitor example when asked about the fourth equation.
• Show the divergence or Stokes theorem step explicitly when converting integral to point form.
• Keep the syllabus to Cartesian coordinates only; no cylindrical or spherical forms are needed.
• In Faraday's law numericals keep the minus sign and explain it with Lenz's law.

FAQs

What is the difference between E and D, and between B and H?

E (V/m) and B (tesla) are the fields that exert forces on charges. D = εE (C/m²) and H = B/μ (A/m) are auxiliary fields whose sources are only free charges and free currents, which makes Maxwell's equations in matter simpler.

Why does the second Maxwell equation have zero on the right side?

Because no isolated magnetic poles have ever been found; every magnet has north and south together, so magnetic field lines always close on themselves and the net flux out of any closed surface is zero.

Is displacement current a real flow of charge?

No. It is a changing electric field, but it produces a magnetic field exactly as a conduction current would, which is why it is called a current.

Why are Maxwell's equations important to engineers?

They govern motors, generators, transformers, antennas, waveguides, optical fibres and wireless communication; every electromagnetic device is designed using them.

These notes follow the University of Mumbai first-year engineering syllabus (NEP 2020, Applied Physics) and were prepared with AI help and checked.