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Practice MCQ

Unit 1 - Notes

PHY109

Unit 1: Electromagnetic Theory

1. Fundamentals of Vector Calculus

To understand electromagnetic theory, one must first grasp the mathematical framework used to describe fields.

Scalar and Vector Fields

Scalar Field: A region in space where a scalar quantity varies from point to point. It is defined by a single magnitude at every point.
- Examples: Temperature distribution $T(x,y,z)$ , Electric Potential $V(x,y,z)$ .
Vector Field: A region in space where a vector quantity implies both magnitude and direction at every point.
- Examples: Velocity of fluid flow $\vec{v}(x,y,z)$ , Electric Field $\vec{E}(x,y,z)$ , Magnetic Field $\vec{B}(x,y,z)$ .

The Del Operator ( $\nabla$ )

The vector differential operator, denoted by nabla ( $\nabla$ ), is defined in Cartesian coordinates as:

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

2. Gradient, Divergence, and Curl

Gradient (Scalar $\to$ Vector)

The gradient of a scalar field $\phi$ represents the maximum rate of change of that scalar function in space.

Mathematical Form:
$\text{grad } \phi = \nabla \phi = \hat{i}\frac{\partial \phi}{\partial x} + \hat{j}\frac{\partial \phi}{\partial y} + \hat{k}\frac{\partial \phi}{\partial z}$
Physical Significance:
- The direction of $\nabla \phi$ is normal (perpendicular) to the level surface (equipotential surface).
- It points in the direction of the steepest increase of $\phi$ .
- Relation to Force: Conservative forces are negative gradients of potential energy ( $\vec{F} = -\nabla V$ ).

Divergence (Vector $\to$ Scalar)

Divergence measures how much a vector field spreads out (diverges) from a point.

Mathematical Form:
$\text{div } \vec{A} = \nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$
Physical Significance:
- Positive Divergence: The point is a source (flux is leaving the volume).
- Negative Divergence: The point is a sink (flux is entering the volume).
- Zero Divergence: The field is Solenoidal (incompressible flow; whatever enters, leaves). Magnetic fields are always solenoidal ( $\nabla \cdot \vec{B} = 0$ ).

Curl (Vector $\to$ Vector)

Curl measures the rotation or "circulation" of a vector field around a point.

Mathematical Form:
$\text{curl } \vec{A} = \nabla \times \vec{A} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}$
Physical Significance:
- It represents the angular velocity of the field at a point.
- If $\nabla \times \vec{A} = 0$ , the field is Irrotational (or Conservative). Electrostatic fields are irrotational ( $\nabla \times \vec{E} = 0$ ) in the absence of changing magnetic fields.

3. Integral Theorems (Qualitative)

These theorems relate differential (point-wise) operations to integral (macroscopic) operations.

Gauss’s Divergence Theorem

Statement: The surface integral of the normal component of a vector function $\vec{A}$ taken over a closed surface $S$ is equal to the volume integral of the divergence of $\vec{A}$ taken over the volume $V$ enclosed by the surface.
Mathematical Form:
$\oint_S \vec{A} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{A}) \, dV$
Utility: Converts a surface integral into a volume integral.

Stokes’ Theorem

Statement: The line integral of the tangential component of a vector function $\vec{A}$ around a simple closed curve $C$ is equal to the surface integral of the curl of $\vec{A}$ taken over any open surface $S$ bounded by the curve.
Mathematical Form:
$\oint_C \vec{A} \cdot d\vec{l} = \int_S (\nabla \times \vec{A}) \cdot d\vec{S}$
Utility: Converts a line integral into a surface integral.

4. Poisson and Laplace Equations

These are second-order partial differential equations derived from Gauss's Law for electrostatics.

Gauss's Law (Differential form): $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$
Relation between Field and Potential: $\vec{E} = -\nabla V$

Substituting (2) into (1):

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

Poisson’s Equation

Used when a charge distribution $\rho$ exists in the region.

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

Laplace’s Equation

Used in a charge-free region ( $\rho = 0$ ).

\nabla^2 V = 0

Significance: Solutions to Laplace's equation are called harmonic functions. It is crucial for solving boundary value problems in electrostatics, heat flow, and fluid dynamics.

5. Equation of Continuity

The continuity equation represents the Law of Conservation of Charge. It states that charge cannot be created or destroyed, only moved.

Derivation Logic:
The total current $I$ flowing out of a closed surface $S$ must equal the rate of decrease of the charge $q$ enclosed by that surface.

I = \oint_S \vec{J} \cdot d\vec{S} = -\frac{dq}{dt}

Since $q = \int_V \rho \, dV$ :

\oint_S \vec{J} \cdot d\vec{S} = -\frac{\partial}{\partial t} \int_V \rho \, dV

Using Gauss's Divergence Theorem on the left side:

\int_V (\nabla \cdot \vec{J}) \, dV = \int_V \left(-\frac{\partial \rho}{\partial t}\right) \, dV

Final Equation

\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 Modification

Ampere’s Circuital Law (Original)

Statement: The line integral of the magnetic field $\vec{B}$ around any closed path is equal to $\mu_0$ times the net steady current $I_{enc}$ enclosed by the path.

\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 equation:
$\nabla \cdot (\nabla \times \vec{B}) = \mu_0 (\nabla \cdot \vec{J})$
Vector identity states divergence of a curl is always zero: $0 = \mu_0 (\nabla \cdot \vec{J})$ .
This implies $\nabla \cdot \vec{J} = 0$ , which is only true for steady currents.
However, the continuity equation states $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$ . If charge density changes (like in a charging capacitor), Ampere’s law fails.

Maxwell’s Displacement Current

To fix this, Maxwell introduced the Displacement Current Density ( $\vec{J}_d$ ).
He modified the current term: $\vec{J}_{total} = \vec{J}_{conduction} + \vec{J}_{displacement}$ .

Using Gauss's Law ( $\nabla \cdot \vec{D} = \rho$ , where $\vec{D} = \epsilon_0 \vec{E}$ ):

\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial t} (\nabla \cdot \vec{D}) = -\nabla \cdot \frac{\partial \vec{D}}{\partial t}

\nabla \cdot \left( \vec{J} + \frac{\partial \vec{D}}{\partial t} \right) = 0

Thus, the displacement current density is defined as:

\vec{J}_d = \frac{\partial \vec{D}}{\partial t} = \epsilon_0 \frac{\partial \vec{E}}{\partial t}

Modified Ampere’s Law (Ampere-Maxwell Law)

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

Significance: This showed that a changing electric field produces a magnetic field, just like a conduction current does.

7. Maxwell’s Electromagnetic Equations

Maxwell unified electricity and magnetism into a set of four fundamental equations.

1. Gauss’s Law for Electrostatics

Differential: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$
Integral: $\oint_S \vec{E} \cdot d\vec{S} = \frac{q_{enc}}{\epsilon_0}$
Physical Significance: Electric field lines diverge from positive charges and converge on negative charges. It relates the electric field to the charge distribution.

2. Gauss’s Law for Magnetism

Differential: $\nabla \cdot \vec{B} = 0$
Integral: $\oint_S \vec{B} \cdot d\vec{S} = 0$
Physical Significance:
- Magnetic monopoles do not exist (isolated North or South poles are impossible).
- Magnetic field lines are continuous closed loops; they have no starting or ending point.

3. Faraday’s Law of Electromagnetic Induction

Differential: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
Integral: $\oint_C \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
Physical Significance:
- A time-varying magnetic field generates an electric field.
- This is the principle behind generators and transformers.
- The electric field here is non-conservative.

4. Ampere-Maxwell Law

Differential: $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$
Integral: $\oint_C \vec{B} \cdot d\vec{l} = \mu_0 (I_{enc} + I_d)$
Physical Significance:
- Magnetic fields are generated by two sources:
  1. Conduction current ( $\vec{J}$ ).
  2. Changing electric fields (Displacement current, $\partial \vec{E}/\partial t$ ).
- This equation predicts the existence of electromagnetic waves.

Summary of Maxwell's Equations (Free Space)

In free space (vacuum), $\rho = 0$ and $\vec{J} = 0$ . The equations become:

$\nabla \cdot \vec{E} = 0$
$\nabla \cdot \vec{B} = 0$
$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
$\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$

These equations explicitly show that a changing E-field creates a B-field, and a changing B-field creates an E-field, allowing electromagnetic waves to propagate through a vacuum.