GATE Electrical Engineering Electromagnetic Fields Syllabus

What is the electric field just outside a conductor carrying surface charge $\rho_{s}$?

$$\vec{E} = \frac{\rho_{s}}{\varepsilon_{0}}\,\hat{a}_{n}, \qquad \vec{D} = \rho_{s}\,\hat{a}_{n}$$ directed normal to the conductor surface; inside the conductor $\vec{E}=0$.

Using Gauss's law, what is $\vec{E}$ inside ($r<a$) a uniformly charged solid sphere of radius $a$ and total charge $Q$?

$$\vec{E} = \frac{Q\,r}{4\pi\varepsilon_{0} a^{3}}\,\hat{a}_{r}$$ The field grows linearly with $r$ inside the sphere.

What is $\vec{E}$ outside ($r>a$) a uniformly charged sphere of total charge $Q$?

$$\vec{E} = \frac{Q}{4\pi\varepsilon_{0} r^{2}}\,\hat{a}_{r}$$ Same as if all charge were concentrated at the centre.

Relate electric field intensity to electric potential $V$.

$$\vec{E} = -\nabla V$$ The field is the negative gradient of potential, pointing in the direction of greatest potential decrease.

How is potential difference obtained from the electric field?

$$V_{AB} = V_{A} - V_{B} = -\int_{B}^{A} \vec{E}\cdot d\vec{l}$$ The line integral of $\vec{E}$ gives the work per unit charge.

What does it mean that the electrostatic field is conservative?

The closed line integral of $\vec{E}$ is zero, $\oint \vec{E}\cdot d\vec{l} = 0$, equivalently $\nabla\times\vec{E} = 0$; work done is path-independent.

State the differential form of Gauss's law for $\vec{E}$ in free space.

$$\nabla\cdot\vec{E} = \frac{\rho_{v}}{\varepsilon_{0}}$$

What is the force on a charge $q$ placed in an electric field $\vec{E}$?

$$\vec{F} = q\vec{E}$$ For a positive charge the force is along $\vec{E}$; for a negative charge it is opposite to $\vec{E}$.

Compare the directions of $\vec{E}$ relative to a positive and a negative point charge.

For a positive charge $\vec{E}$ points radially outward (away from the charge); for a negative charge $\vec{E}$ points radially inward (toward the charge).

Distinguish $\vec{E}$ and $\vec{D}$ in terms of medium dependence and units.

$\vec{E}$ (units $\text{V/m}$) depends on the medium via $\varepsilon$; $\vec{D}$ (units $\text{C/m}^{2}$) depends only on free charge and is medium-independent. They relate by $\vec{D}=\varepsilon\vec{E}$.

Define electric dipole moment and the field of a dipole far away.

Dipole moment $\vec{p}=Q\vec{d}$ (from $-Q$ to $+Q$). Far-field magnitude $E = \dfrac{p}{4\pi\varepsilon_{0} r^{3}}\sqrt{1+3\cos^{2}\theta}$, varying as $\dfrac{1}{r^{3}}$.

What is the torque on an electric dipole in a uniform field $\vec{E}$?

$$\vec{\tau} = \vec{p}\times\vec{E}, \qquad |\vec{\tau}| = pE\sin\theta$$ tending to align the dipole with the field.

What is the relation between the volume charge density and $\vec{E}$ via the Poisson equation?

$$\nabla^{2} V = -\frac{\rho_{v}}{\varepsilon}$$ Poisson's equation; in a charge-free region it reduces to Laplace's equation $\nabla^{2}V = 0$.

What is the electric flux density on a Gaussian sphere of radius $r$ enclosing charge $Q$, and why is it uniform?

$D = \dfrac{Q}{4\pi r^{2}}$. By spherical symmetry $\vec{D}$ is radial and constant in magnitude over the surface, so $\oint \vec{D}\cdot d\vec{S} = D\cdot 4\pi r^{2} = Q$.

State Coulomb's law for the force between two point charges $Q_{1}$ and $Q_{2}$.

$$\vec{F} = \frac{Q_{1}Q_{2}}{4\pi\varepsilon_{0} R^{2}}\,\hat{a}_{R}$$ Like charges repel, unlike charges attract.

How are field lines of $\vec{E}$ related to flux lines of $\vec{D}$?

They have the same direction at every point. $\vec{D}$ lines (flux lines) begin on positive charge and end on negative charge; the number of $\vec{D}$ lines from a charge equals $Q$.

What is the electric field intensity inside a conductor in electrostatic equilibrium?

It is zero ($\vec{E}=0$). All net charge resides on the surface, and the conductor is an equipotential body.

What is the tangential component of $\vec{E}$ at the boundary between two dielectrics?

It is continuous: $E_{t1} = E_{t2}$ (assuming no surface current). This follows from $\oint \vec{E}\cdot d\vec{l}=0$.

What is the normal component condition for $\vec{D}$ at a dielectric boundary with free surface charge $\rho_{s}$?

$$D_{n1} - D_{n2} = \rho_{s}$$ If no free surface charge, $D_{n1}=D_{n2}$ (normal component of $\vec{D}$ is continuous).

Give the energy density stored in an electrostatic field.

$$w_{E} = \frac{1}{2}\vec{D}\cdot\vec{E} = \frac{1}{2}\varepsilon E^{2} = \frac{D^{2}}{2\varepsilon}\quad (\text{J/m}^{3})$$

How does the field of an infinite line charge differ from that of a point charge in distance dependence?

A point charge field varies as $\dfrac{1}{r^{2}}$, while an infinite line charge field varies as $\dfrac{1}{\rho}$ (inverse first power), and an infinite sheet field is independent of distance.

What is the total flux leaving a closed surface enclosing zero net charge?

Zero. By Gauss's law $\oint_{S}\vec{D}\cdot d\vec{S} = Q_{\text{enc}} = 0$; flux entering equals flux leaving.

Define surface charge density $\rho_{s}$ and volume charge density $\rho_{v}$.

$\rho_{s} = \dfrac{dQ}{dS}$ (charge per unit area, $\text{C/m}^{2}$) and $\rho_{v} = \dfrac{dQ}{dv}$ (charge per unit volume, $\text{C/m}^{3}$).

Write the electric field of a uniformly charged ring of radius $a$, charge $Q$, at axial distance $z$.

$$\vec{E} = \frac{Q z}{4\pi\varepsilon_{0}\,(z^{2}+a^{2})^{3/2}}\,\hat{a}_{z}$$ directed along the axis; it is zero at the centre ($z=0$).

For a charge enclosed by a closed surface, why is $\oint \vec{D}\cdot d\vec{S}$ independent of the surface shape?

Because Gauss's law depends only on the enclosed charge, not the geometry: $\oint \vec{D}\cdot d\vec{S} = Q_{\text{enc}}$ for any closed surface enclosing $Q$.

State the relationship $\nabla\cdot\vec{D}=\rho_v$ derived from the integral Gauss's law using which theorem?

The divergence (Gauss's) theorem: $\oint_{S}\vec{D}\cdot d\vec{S} = \int_{v}(\nabla\cdot\vec{D})\,dv$, equated to $\int_{v}\rho_{v}\,dv$ gives $\nabla\cdot\vec{D}=\rho_{v}$.

What is the magnitude of $\vec{E}$ midway between two equal positive charges, and why?

It is zero at the midpoint because the two fields are equal in magnitude and opposite in direction, canceling each other.

How is electric flux density used to define the dielectric polarization relation $\vec{D}$?

$$\vec{D} = \varepsilon_{0}\vec{E} + \vec{P}$$ where $\vec{P}$ is the polarization (dipole moment per unit volume); in linear media $\vec{P}=\varepsilon_{0}\chi_{e}\vec{E}$ and $\varepsilon_{r}=1+\chi_{e}$.

At a field point, how do you find the unit vector $\hat{a}_{R}$ used in field formulas?

$$\hat{a}_{R} = \frac{\vec{R}}{|\vec{R}|} = \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|}$$ where $\vec{r}$ is the field point and $\vec{r}'$ the source point.

Summarize the inverse relationship of $E$ with distance for point, line, and sheet charges.

Point charge: $E\propto \dfrac{1}{r^{2}}$; infinite line charge: $E\propto \dfrac{1}{\rho}$; infinite sheet of charge: $E$ is constant (independent of distance).