Unit 1 - Notes

PHY110

Unit 1: Electromagnetic Theory

1. Scalar and Vector Fields

To understand electromagnetism, one must first understand the mathematical framework of fields used to describe physical quantities distributed in space.

Scalar Fields

A scalar field is a region in space where a scalar physical quantity is defined at every point. It is represented by a function $\phi(x, y, z)$ that depends only on position coordinates.

Characteristics: Has magnitude only; no direction.
Examples: Temperature distribution in a rod $T(x,y,z)$ , Electric Potential $V(x,y,z)$ , Density $\rho(x,y,z)$ .
Representation: Often represented by level surfaces (equipotential surfaces or isotherms) where the value of the scalar is constant.

Vector Fields

A vector field is a region in space where a vector physical quantity is defined at every point. It is represented by a vector function $\vec{A}(x, y, z)$ .

Characteristics: Has both magnitude and direction at every point.
Examples: Electric Field Intensity $\vec{E}$ , Magnetic Field Intensity $\vec{H}$ , Velocity of fluid flow $\vec{v}$ .
Representation: Represented by field lines (flux lines/streamlines). The tangent to the line gives the direction, and the density of lines represents the magnitude.

2. Vector Calculus Operators: Gradient, Divergence, and Curl

The Del operator (Nabla), denoted by $\nabla$ , is a vector differential operator defined as:

\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}

Gradient (Grad)

The gradient operates on a scalar field and results in a vector field.

Definition: If $\phi(x,y,z)$ is a scalar function:
$\text{grad } \phi = \nabla \phi = \hat{i}\frac{\partial \phi}{\partial x} + \hat{j}\frac{\partial \phi}{\partial y} + \hat{k}\frac{\partial \phi}{\partial z}$
Physical Significance:
1. Represents the maximum rate of change of the scalar function with respect to space.
2. The direction of the gradient is normal (perpendicular) to the level surface ( $\phi = \text{constant}$ ).
3. Example in EM: $\vec{E} = -\nabla V$ (Electric field is the negative gradient of electric potential).

Divergence (Div)

The divergence operates on a vector field and results in a scalar field. It involves the dot product with $\nabla$ .

Definition: If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ :
$\text{div } \vec{A} = \nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$
Physical Significance:
1. It measures the net outward flux of a vector field per unit volume.
2. Positive Divergence: The point is a source (flux diverges/leaves).
3. Negative Divergence: The point is a sink (flux converges/enters).
4. Zero Divergence ( $\nabla \cdot \vec{A} = 0$ ): The vector field is solenoidal (incompressible; what goes in must come out).

Curl

The curl operates on a vector field and results in a vector field. It involves the cross product with $\nabla$ .

Definition:
$\text{curl } \vec{A} = \nabla \times \vec{A} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}$
Physical Significance:
1. It measures the rotation or angular velocity of the field at a point (circulation per unit area).
2. If $\nabla \times \vec{A} \neq 0$ , the field is rotational (contains vortices).
3. If $\nabla \times \vec{A} = 0$ , the field is irrotational (conservative). Electrostatic fields are irrotational.

3. Fundamental Integral Theorems

Gauss’s Divergence Theorem

Relates a Volume Integral to a Surface Integral.

Statement: The volume integral of the divergence of a vector field $\vec{A}$ taken over a volume $V$ is equal to the surface integral of the normal component of $\vec{A}$ over the closed surface $S$ enclosing that volume.
Mathematical Form:
$\iiint_V (\nabla \cdot \vec{A}) dV = \oiint_S \vec{A} \cdot d\vec{S}$
Qualitative Meaning: The total amount of "stuff" being generated inside a box (sum of sources/sinks) is exactly equal to the amount of "stuff" flowing out through the walls of the box.

Stokes’ Theorem

Relates a Surface Integral to a Line Integral.

Statement: The surface integral of the curl of a vector field $\vec{A}$ over an open surface $S$ is equal to the line integral of the vector $\vec{A}$ around the closed contour $C$ bounding that surface.
Mathematical Form:
$\iint_S (\nabla \times \vec{A}) \cdot d\vec{S} = \oint_C \vec{A} \cdot d\vec{l}$
Qualitative Meaning: The sum of all the "swirling" (micro-rotations) inside a loop adds up to the total circulation along the boundary of the loop.

4. Poisson and Laplace Equations

These equations are derived from Gauss's Law for electrostatics ( $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ ) and the definition of potential ( $\vec{E} = -\nabla V$ ).

Substituting $\vec{E}$ into Gauss's Law:

\nabla \cdot (-\nabla V) = \frac{\rho}{\epsilon_0} \implies -\nabla^2 V = \frac{\rho}{\epsilon_0}

Poisson’s Equation

\nabla^2 V = -\frac{\rho}{\epsilon_0}

Describes the potential distribution in a region where charge density $\rho$ exists.
Used to find the potential $V$ if the charge distribution is known.

Laplace’s Equation

If the region is charge-free (i.e., $\rho = 0$ ), Poisson’s equation reduces to:

\nabla^2 V = 0

Describes the potential in free space or between conductors.
It is a partial differential equation essential for solving electrostatic boundary value problems.

5. Continuity Equation

The continuity equation represents the law of conservation of charge. It states that charge can neither be created nor destroyed, only moved.

Concept: The current $I$ flowing out of a closed surface must equal the rate of decrease of charge $Q$ inside that surface.
$I = -\frac{dQ}{dt}$
Derivation:
Using $I = \oint \vec{J} \cdot d\vec{S}$ and $Q = \int \rho dV$ :
$\oint_S \vec{J} \cdot d\vec{S} = -\frac{d}{dt} \iiint_V \rho dV$
Applying Gauss's Divergence Theorem to the LHS:
$\iiint_V (\nabla \cdot \vec{J}) dV = -\iiint_V \frac{\partial \rho}{\partial t} dV$
Final Form:
$\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0$
Steady Currents: For steady currents, charge density does not change with time ( $\frac{\partial \rho}{\partial t} = 0$ ), so $\nabla \cdot \vec{J} = 0$ .

6. Ampere’s Circuital Law and Maxwell’s Correction

Ampere’s Circuital Law (Original)

It states that the line integral of the magnetic field $\vec{B}$ around a closed path is equal to $\mu_0$ times the net current $I_{enc}$ enclosed by the path.

Integral Form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
Differential Form: $\nabla \times \vec{B} = \mu_0 \vec{J}$

The Inconsistency

Maxwell noticed a flaw in Ampere's law for time-varying fields.

Take the divergence of Ampere's differential form: $\nabla \cdot (\nabla \times \vec{B}) = \nabla \cdot (\mu_0 \vec{J})$ .
Vector identity states divergence of a curl is always zero: $\nabla \cdot (\nabla \times \vec{B}) = 0$ .
Therefore, Ampere’s law implies $\nabla \cdot \vec{J} = 0$ .
However, the continuity equation states $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$ .
Conclusion: Ampere's original law is only valid for steady currents ( $\frac{\partial \rho}{\partial t} = 0$ ), not time-varying ones (like a charging capacitor).

Maxwell’s Displacement Current

To fix this, Maxwell added a term to Ampere's Law. He proposed that a changing electric field produces a magnetic field, just like a conduction current does.

Displacement Current Density ( $\vec{J}_d$ ):
$\vec{J}_d = \epsilon_0 \frac{\partial \vec{E}}{\partial t} = \frac{\partial \vec{D}}{\partial t}$
This is not a flow of physical charge, but a time-varying electric field acting as a current.

Modified Ampere’s Law

Total current density is the sum of Conduction Current ( $\vec{J}_c$ ) and Displacement Current ( $\vec{J}_d$ ).

\nabla \times \vec{B} = \mu_0 (\vec{J}_c + \vec{J}_d)

\nabla \times \vec{B} = \mu_0 \left( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right)

7. Maxwell’s Electromagnetic Equations

Maxwell’s equations are a set of four coupled partial differential equations that form the foundation of classical electrodynamics, optics, and electric circuits.

Notation: $\vec{E}$ = Electric field, $\vec{B}$ = Magnetic flux density, $\vec{D}$ = Electric displacement ( $\epsilon \vec{E}$ ), $\vec{H}$ = Magnetic field intensity ( $\vec{B}/\mu$ ), $\rho$ = Charge density, $\vec{J}$ = Current density.

The Equations (Table)

Name	Differential Form	Integral Form
1. Gauss’s Law (Electrostatics)	$\nabla \cdot \vec{D} = \rho$	$\oiint_S \vec{D} \cdot d\vec{S} = Q_{enc}$
2. Gauss’s Law (Magnetism)	$\nabla \cdot \vec{B} = 0$	$\oiint_S \vec{B} \cdot d\vec{S} = 0$
3. Faraday’s Law of Induction	$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$	$\oint_C \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
4. Ampere-Maxwell Law	$\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$	$\oint_C \vec{H} \cdot d\vec{l} = I_{enc} + \int_S \frac{\partial \vec{D}}{\partial t} \cdot d\vec{S}$

8. Physical Significance of Maxwell's Equations

1. Maxwell’s First Equation ( $\nabla \cdot \vec{D} = \rho$ )

Significance: Electric field lines originate from positive charges and terminate on negative charges.
It relates the spatial variation of the electric field to the charge density.
It represents Coulomb’s law in a generalized form.

2. Maxwell’s Second Equation ( $\nabla \cdot \vec{B} = 0$ )

Significance: Magnetic field lines are continuous closed loops; they have no starting or ending point.
Non-existence of Monopoles: There are no isolated magnetic poles (no magnetic "charge"). You cannot have a North pole without a South pole.
The net magnetic flux through any closed surface is zero.

3. Maxwell’s Third Equation ( $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ )

Significance: A time-varying magnetic field generates (induces) an electric field.
This is the principle behind electromagnetic induction, transformers, and generators.
The negative sign indicates Lenz’s Law (the induced effect opposes the cause).

4. Maxwell’s Fourth Equation ( $\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$ )

Significance: Magnetic fields are generated by two sources:
1. Conduction current ( $\vec{J}$ - flow of electrons).
2. Displacement current ( $\frac{\partial \vec{D}}{\partial t}$ - time-varying electric field).
This equation predicts the existence of electromagnetic waves, proving that light is an electromagnetic wave by showing that changing $\vec{E}$ creates $\vec{B}$ , and changing $\vec{B}$ creates $\vec{E}$ , allowing fields to propagate through space.