Engineering Physics (BT-201) Unit 5 Notes

How unit 5 is examined

This unit covers field and potential of charge distributions, dielectrics, vector calculus theorems, the continuity equation and Maxwell's equations; the continuity equation carries the most marks, then field and potential, Stokes' theorem and Maxwell's equations.

Calculation of electric field and electrostatic potential for a charge distribution

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Definition. Electric field intensity is the force per unit positive test charge, $\vec{E}=\vec{F}/q_0$ (unit N/C or V/m). Electrostatic potential at a point is the work done per unit charge in bringing a test charge from infinity to that point, $V=-\int_\infty^{r}\vec{E}\cdot d\vec{l}$.

Key points.

Charge densities are linear $\lambda=dq/dl$, surface $\sigma=dq/da$ and volume $\rho=dq/dV$, so the element is $dq=\lambda\,dl$, $\sigma\,da$ or $\rho\,dV$.
A point charge gives $\vec{E}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}\hat{r}$ and $V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$.
By superposition, a volume distribution gives $\vec{E}(\vec{r})=\dfrac{1}{4\pi\varepsilon_0}\displaystyle\int_V \dfrac{\rho(\vec{r}')(\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3}dV'$.
Likewise the potential is a scalar sum, $V(\vec{r})=\dfrac{1}{4\pi\varepsilon_0}\displaystyle\int_V \dfrac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}dV'$, so it is easier to compute than $\vec{E}$.
Field and potential are related by $\vec{E}=-\nabla V$, because $dV=-\vec{E}\cdot d\vec{l}$ and $\nabla\times\vec{E}=0$ in electrostatics.

Infinite line charge (Gauss's law). Take a coaxial cylinder of radius $r$ and length $l$ around a line of density $\lambda$. By symmetry $\vec{E}$ is radial, so the flux through the two end caps is zero and $E$ is constant on the curved surface.

$$\oint \vec{E}\cdot d\vec{S}=E(2\pi r l)=\frac{q_{enc}}{\varepsilon_0}=\frac{\lambda l}{\varepsilon_0}$$

$$\Rightarrow\; E=\frac{\lambda}{2\pi\varepsilon_0 r}$$

==The electric field of an infinite line charge is $E=\lambda/(2\pi\varepsilon_0 r)$, directed radially outward and falling as $1/r$.==

Answer frame. Open with the definition of $\vec{E}$ and its unit; draw the line charge with the coaxial cylindrical Gaussian surface of radius $r$, length $l$ and flux arrows on the curved side only; show caps contribute zero, apply Gauss's law and close with $E=\lambda/2\pi\varepsilon_0 r$. For the general derivation, define the three densities, integrate $d\vec{E}$, then give $V$ and end with $\vec{E}=-\nabla V$. For the 5-mark potential question write points 1, 2, 4 and the line, surface and volume forms.

Asked: [7 marks] (Jun 2023) Define the electric field intensity. Find the expression for electric intensity due to an infinite line charge. Asked: [7 marks] (Dec 2024) Derive the electric field and electrostatic potential for a charge distribution. Asked: [5 marks] (Dec 2023) Explain the electrostatic potential for a charge distribution.

Electric displacement

Definition. Electric displacement $\vec{D}=\varepsilon_0\vec{E}+\vec{P}$ is a field that depends only on free charge, where $\vec{P}$ is the polarization.

Key points.

For a linear isotropic dielectric $\vec{P}=\varepsilon_0\chi_e\vec{E}$, so $\vec{D}=\varepsilon_0\varepsilon_r\vec{E}=\varepsilon\vec{E}$.
Gauss's law in a dielectric becomes $\oint\vec{D}\cdot d\vec{S}=q_{free}$, or $\nabla\cdot\vec{D}=\rho_f$.
In vacuum $\vec{P}=0$, so $\vec{D}=\varepsilon_0\vec{E}$; its unit is C/m$^2$.

Basic Introduction to Dielectrics

Definition. A dielectric is an insulating material with no free charges that becomes polarized in an electric field; polarization $\vec{P}$ is the dipole moment per unit volume, $\vec{P}=N\vec{p}=N\alpha\vec{E}_{local}$.

Key points.

In an applied field the bound charges of a dielectric shift slightly, creating induced dipoles that oppose the applied field inside and raise capacitance by the factor $\varepsilon_r$.
Electronic polarization is the displacement of the electron cloud relative to the nucleus, occurs in all materials and is independent of temperature.
Ionic polarization is the relative displacement of positive and negative ions in ionic solids such as NaCl and is also independent of temperature.
Orientational (dipolar) polarization is the alignment of permanent dipoles, as in water or HCl, and decreases as temperature rises because thermal agitation disorders them.
Space charge polarization is the piling up of charges at interfaces or grain boundaries in a non-uniform dielectric and appears only at low frequency.
Total polarizability is $\alpha=\alpha_e+\alpha_i+\alpha_o+\alpha_s$; at optical frequencies only $\alpha_e$ survives.

==Polarization is the dipole moment per unit volume induced in a dielectric, and the total polarizability is $\alpha=\alpha_e+\alpha_i+\alpha_o+\alpha_s$.==

Answer frame. Open with the definition of a dielectric and $\vec{P}$; draw a small before/after sketch of an atom in a field; develop the four types in the order electronic, ionic, orientational, space charge with temperature and frequency behaviour of each; close with the total polarizability.

Asked: [7 marks] (Jun 2022, Jun 2025) Write a note on different types of polarization in dielectric materials.

Gradient, Divergence and curl

Definition. Gradient of a scalar $\phi$ is $\nabla\phi=\hat{i}\frac{\partial\phi}{\partial x}+\hat{j}\frac{\partial\phi}{\partial y}+\hat{k}\frac{\partial\phi}{\partial z}$; divergence of a vector $\vec{A}$ is $\nabla\cdot\vec{A}=\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$; curl is $\nabla\times\vec{A}$, the determinant of $\hat{i},\hat{j},\hat{k}$, $\partial/\partial x,y,z$ and $A_x,A_y,A_z$.

Key points.

The gradient is a vector pointing in the direction of steepest increase of $\phi$, with magnitude equal to that maximum rate, and it is normal to equipotential surfaces.
Divergence is a scalar giving the net outward flux per unit volume, so positive divergence marks a source and negative marks a sink.
Curl is a vector measuring the rotation or circulation per unit area of the field.
Electrostatics uses $\vec{E}=-\nabla V$, $\nabla\cdot\vec{E}=\rho/\varepsilon_0$ and $\nabla\times\vec{E}=0$.

Asked: [5 marks] (Dec 2023) Define gradient of a scalar field, divergence of a vector field.

Stokes' theorem

Statement. The line integral of a vector field around a closed path $C$ equals the surface integral of its curl over any open surface $S$ bounded by $C$:

$$\oint_C \vec{A}\cdot d\vec{l}=\iint_S(\nabla\times\vec{A})\cdot d\vec{S}$$

Proof.

Divide $S$ into small rectangular loops of area $dx\,dy$ in the $xy$-plane.
Circulation around one loop, taking the sides in order, is $A_x dx+\left(A_y+\frac{\partial A_y}{\partial x}dx\right)dy-\left(A_x+\frac{\partial A_x}{\partial y}dy\right)dx-A_y dy$.
This simplifies to $\left(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}\right)dx\,dy=(\nabla\times\vec{A})_z\,dx\,dy$.
Adding over all loops, each interior side is traversed twice in opposite directions and cancels, leaving only the outer boundary $C$ on the left.
The right side sums to $\iint(\nabla\times\vec{A})\cdot d\vec{S}$, which proves the theorem.

<mark>Stokes' theorem converts the closed line integral of $\vec{A}$ into the surface integral of $\nabla\times\vec{A}$ over the surface bounded by that path.</mark>

Answer frame. Open with the statement; draw the open surface split into small loops with the boundary curve $C$ and arrows showing cancelling interior sides; do steps 1-5 in order; close with the equation. For the "Gauss and Stokes" questions add the Gauss theorem (next but two sections) and the physical significance of each: Stokes links circulation to curl, Gauss links outward flux to divergence.

Asked: [4 marks] (Dec 2023) State and prove Stokes' theorem. Asked: [7 marks] (Dec 2024) State and derive the divergence theorem (Gauss) and Stokes' theorem. Asked: [7 marks] (Nov 2022) State and explain Stokes' and Gauss theorem.

Gauss Theorem

Definition. The divergence theorem states $\oint_S\vec{A}\cdot d\vec{S}=\int_V(\nabla\cdot\vec{A})\,dV$: outward flux through a closed surface equals the volume integral of the divergence inside.

Key points.

Proof: split $V$ into small cubes; the outward flux of one cube is $(\nabla\cdot\vec{A})\,dx\,dy\,dz$.
Adding cubes, flux through shared faces cancels, leaving only the outer surface $S$.
Significance: it turns a surface integral into a volume integral, and gives $\nabla\cdot\vec{E}=\rho/\varepsilon_0$ from Gauss's law.

Continuity equation for current densities

Definition. The continuity equation expresses conservation of charge: the current leaving a volume equals the rate of decrease of charge inside it, $\nabla\cdot\vec{J}+\dfrac{\partial\rho}{\partial t}=0$.

Derivation.

Charge can neither be created nor destroyed, so any charge leaving a closed surface $S$ must come from inside its volume $V$.
The total current leaving $S$ is $I=\oint_S\vec{J}\cdot d\vec{S}$, where $\vec{J}$ is the current density in A/m$^2$.
The charge inside is $q=\int_V\rho\,dV$, so its rate of decrease is $-\dfrac{d}{dt}\int_V\rho\,dV=-\int_V\dfrac{\partial\rho}{\partial t}dV$.
Equating: $\oint_S\vec{J}\cdot d\vec{S}=-\int_V\dfrac{\partial\rho}{\partial t}dV$.
Apply the divergence theorem to the left side: $\int_V(\nabla\cdot\vec{J})\,dV=-\int_V\dfrac{\partial\rho}{\partial t}dV$.
The volume is arbitrary, so the integrands are equal:

$$\nabla\cdot\vec{J}=-\frac{\partial\rho}{\partial t}\quad\Longleftrightarrow\quad\nabla\cdot\vec{J}+\frac{\partial\rho}{\partial t}=0$$

Key points.

Steady state has $\partial\rho/\partial t=0$, so $\nabla\cdot\vec{J}=0$: as much current enters a region as leaves it, and the current lines are closed.
The equation is the differential form of charge conservation and holds in every medium, since it follows from the definitions of $\vec{J}$ and $\rho$ alone.
Assumptions: charge is conserved locally and the volume is fixed in space.
Combined with Ampere's law it forces the displacement current term $\varepsilon_0\partial\vec{E}/\partial t$ into Maxwell's fourth equation, since $\nabla\cdot(\nabla\times\vec{H})=0$.

==The continuity equation $\nabla\cdot\vec{J}=-\partial\rho/\partial t$ states that charge is conserved: net current out of a point equals the rate of fall of charge density there.==

Answer frame. Open with the law of conservation of charge; draw a closed surface $S$ enclosing volume $V$ with $\vec{J}$ arrows leaving it; develop steps 1-6 in order, naming the divergence theorem in step 5; close with the differential form and the steady-state case $\nabla\cdot\vec{J}=0$ as its significance.

Asked: [7 marks] (Jun 2022, Jun 2023, Dec 2024, Jun 2025) Discuss the continuity equation. State and prove the equation of continuity and explain its significance; derive it from charge conservation stating assumptions.

Maxwell's equation in vacuum and non-conducting medium

Definition. Maxwell's four equations describe all classical electromagnetism. In vacuum there are no free charges or currents, so $\rho=0$, $\vec{J}=0$, $\vec{D}=\varepsilon_0\vec{E}$ and $\vec{B}=\mu_0\vec{H}$.

General form (differential). $\nabla\cdot\vec{D}=\rho$; $\nabla\cdot\vec{B}=0$; $\nabla\times\vec{E}=-\partial\vec{B}/\partial t$; $\nabla\times\vec{H}=\vec{J}+\partial\vec{D}/\partial t$.

Vacuum form.

No.	Equation	Meaning
1	$\nabla\cdot\vec{E}=0$	Gauss's law: no free charge, so field lines have no source or sink
2	$\nabla\cdot\vec{B}=0$	No magnetic monopoles; magnetic lines are closed loops
3	$\nabla\times\vec{E}=-\dfrac{\partial\vec{B}}{\partial t}$	Faraday's law: a changing magnetic field induces an electric field
4	$\nabla\times\vec{B}=\mu_0\varepsilon_0\dfrac{\partial\vec{E}}{\partial t}$	Ampere-Maxwell law: a changing electric field produces a magnetic field

Key points.

Setting $\rho=0$, $\vec{J}=0$ in the general equations and using the constitutive relations gives the table directly.
Equations 3 and 4 couple $\vec{E}$ and $\vec{B}$, so a changing one sustains the other and a wave can travel through empty space.
Non-conducting medium: $\rho=0$, $\vec{J}=0$ again, but $\varepsilon_0,\mu_0$ are replaced by $\varepsilon,\mu$, so equation 4 becomes $\nabla\times\vec{B}=\mu\varepsilon\,\partial\vec{E}/\partial t$.
Wave speed is $1/\sqrt{\mu_0\varepsilon_0}=c\approx3\times10^8$ m/s in vacuum, and $1/\sqrt{\mu\varepsilon}$ in the medium.

==In vacuum Maxwell's equations are $\nabla\cdot\vec{E}=0$, $\nabla\cdot\vec{B}=0$, $\nabla\times\vec{E}=-\partial\vec{B}/\partial t$ and $\nabla\times\vec{B}=\mu_0\varepsilon_0\,\partial\vec{E}/\partial t$.==

Answer frame. Open by writing the four general equations; state $\rho=0$, $\vec{J}=0$ and the constitutive relations; write the four vacuum equations in the table with the law each represents; close with the coupling of $\vec{E}$ and $\vec{B}$ and speed $c$.

Asked: [7 marks] (Nov 2022) Derive Maxwell equations in vacuum. Asked: [7 marks] (Jun 2025) Explain Maxwell's equations in vacuum.

Poynting vector

Definition. The Poynting vector $\vec{S}=\vec{E}\times\vec{H}$ is the energy flowing per unit time per unit area of an electromagnetic wave, in W/m$^2$.

Key points.

Its direction is that of wave propagation, perpendicular to both $\vec{E}$ and $\vec{H}$.
Poynting's theorem says $-\dfrac{\partial u}{\partial t}=\nabla\cdot\vec{S}+\vec{J}\cdot\vec{E}$: energy leaving a volume plus work done on charges equals the fall in stored field energy.
In vacuum $S=EB/\mu_0=E^2/(\mu_0c)$.

Last-minute revision

$\vec{E}=\vec{F}/q_0$; $\vec{E}=-\nabla V$; $V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}$.
Infinite line charge: $E=\lambda/(2\pi\varepsilon_0 r)$ from a coaxial cylinder; end caps give zero flux.
$\vec{D}=\varepsilon_0\vec{E}+\vec{P}=\varepsilon\vec{E}$, with $\oint\vec{D}\cdot d\vec{S}=q_{free}$.
Polarization types: electronic, ionic, orientational (only this one depends on temperature), space charge (low frequency).
Gradient is a vector, divergence and curl measure source strength and rotation.
Stokes: $\oint\vec{A}\cdot d\vec{l}=\iint(\nabla\times\vec{A})\cdot d\vec{S}$.
Gauss divergence: $\oint\vec{A}\cdot d\vec{S}=\int(\nabla\cdot\vec{A})dV$.
Continuity: $\nabla\cdot\vec{J}+\partial\rho/\partial t=0$; steady state $\nabla\cdot\vec{J}=0$.
Maxwell in vacuum: $\nabla\cdot\vec{E}=0$, $\nabla\cdot\vec{B}=0$, $\nabla\times\vec{E}=-\partial_t\vec{B}$, $\nabla\times\vec{B}=\mu_0\varepsilon_0\partial_t\vec{E}$.
Poynting vector $\vec{S}=\vec{E}\times\vec{H}$; $c=1/\sqrt{\mu_0\varepsilon_0}$.

Memory hooks

Continuity: "Current out = charge down", so the minus sign lives on $\partial\rho/\partial t$.
Polarization order EIOS: Electronic, Ionic, Orientational, Space charge, fastest to slowest.
Divergence theorem is "volume from surface"; Stokes is "surface from edge".
Maxwell in vacuum: two divergences are zero, two curls are cross-coupled (Faraday has a minus sign).

Coverage checklist

Calculation of electric field and electrostatic potential for a charge distribution: Jun 2023 line charge, Dec 2024 derivation, Dec 2023 potential.
Electric displacement: definition, $\vec{D}=\varepsilon_0\vec{E}+\vec{P}$ (no past question).
Basic Introduction to Dielectrics: types of polarization (Jun 2022, Jun 2025).
Gradient, Divergence and curl: Dec 2023 definitions.
Stokes’ theorem: Dec 2023, Dec 2024, Nov 2022.
Gauss Theorem: statement and proof (Dec 2024, Nov 2022 via Stokes section).
Continuity equation for current densities: Jun 2022, Jun 2023, Dec 2024, Jun 2025.
Maxwell’s equation in vacuum and non-conducting medium: Nov 2022, Jun 2025.
Poynting vector: definition and theorem (no past question).

Account

Engineering Physics (BT-201) - Unit 5 Short Notes

How unit 5 is examined

Calculation of electric field and electrostatic potential for a charge distribution

Electric displacement

Basic Introduction to Dielectrics

Gradient, Divergence and curl

Stokes' theorem

Gauss Theorem

Continuity equation for current densities

Maxwell's equation in vacuum and non-conducting medium

Poynting vector

Last-minute revision

Memory hooks

Coverage checklist