Unit I Electromagnetic Theory | PHY 110 Engineering Physics |B.tech

 

1. Scalar and Vector Fields

Scalar Field:

• A scalar field assigns a single value (scalar) to every point in space.
• Example: Temperature in a room can be represented as a scalar field $T(x, y, z)$, where $T$ gives the temperature at coordinates $(x, y, z)$.

Vector Field:

• A vector field assigns a vector to every point in space, representing both magnitude and direction.
• Example: The wind velocity can be expressed as a vector field $\mathbf{V}(x, y, z) = (V_x, V_y, V_z)$, where each component represents the velocity in the respective direction.

2. Gradient, Divergence, and Curl

Gradient ($\nabla f$)

• The gradient of a scalar function $f(x, y, z)$ is a vector field that points in the direction of the greatest increase of $f$. The magnitude of the gradient indicates how steep the increase is.
• Mathematical Definition: $$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$
• Example: Let $f(x, y, z) = x^2 + y^2 + z^2$.
  • Calculate the gradient:
  $$\nabla f = \left( 2x, 2y, 2z \right)$$
  • At the point $(1, 2, 3)$:
  $$\nabla f(1, 2, 3) = (2, 4, 6)$$
  • This means the function increases fastest in the direction of the vector $(2, 4, 6)$.

Divergence ($\nabla \cdot \mathbf{A}$)

• The divergence of a vector field $\mathbf{A} = (A_x, A_y, A_z)$ measures the net rate of flux expansion from a point. It indicates whether the field is expanding (positive divergence) or contracting (negative divergence).
• Mathematical Definition: $$\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$
• Example: For $\mathbf{A} = (x, y, z)$:
  • Calculate the divergence:
  $$\nabla \cdot \mathbf{A} = 1 + 1 + 1 = 3$$
  • This implies a net outflow of field lines at that point.

Curl ($\nabla \times \mathbf{A}$)

• The curl of a vector field measures the tendency of the field to induce rotation around a point.
• Mathematical Definition: $$\nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}, \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}, \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right)$$
• Example: For $\mathbf{A} = (0, 0, z)$:
  • Calculate the curl:
  $$\nabla \times \mathbf{A} = \left( 0 - 0, 0 - 0, 0 - 0 \right) = (0, 0, 0)$$
  • This indicates no rotational tendency in the field.

3. Gauss's Theorem (Divergence Theorem)

Statement: Gauss's Theorem relates the flux of a vector field through a closed surface to the divergence of the field inside the volume bounded by the surface.

• Mathematical Form:

  $$\iint_S \mathbf{A} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{A}) \, dV$$

• Physical Interpretation: The total "outflow" of the vector field through the surface $S$ equals the total source strength (divergence) inside the volume $V$.

• Example: For $\mathbf{A} = \frac{\mathbf{r}}{r^3}$ (where $\mathbf{r}$ is the position vector):

  • Calculate the left side (flux through a sphere of radius $R$):
  $$\iint_S \mathbf{A} \cdot d\mathbf{S} = \int_0^{2\pi} \int_0^{\pi} \frac{1}{R^2} R^2 \sin \theta \, d\theta \, d\phi = 4\pi$$
  • The right side involves finding $\nabla \cdot \mathbf{A}$ (which is zero for $r \neq 0$), thus validating the theorem.

4. Stokes's Theorem

Statement: Stokes's Theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the field over the surface bounded by the curve.

• Mathematical Form:

  $$\oint_C \mathbf{A} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S}$$

• Physical Interpretation: The circulation of the vector field around the curve $C$ is equal to the total curl (or rotation) over the surface $S$.

• Example: For $\mathbf{A} = (0, 0, z)$, find the line integral around a circular path in the xy-plane:

  • The left side calculates the line integral, and the right side finds the curl:
  $$\nabla \times \mathbf{A} = (0, 0, 0) \Rightarrow \text{The integral is } 0.$$

5. Poisson and Laplace Equations

Poisson's Equation

• Statement: Describes the relationship between a scalar potential and its source:

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$$

• Physical Interpretation: The Laplacian of the potential $\phi$ is proportional to the charge density $\rho$.

• Example: For a point charge $q$ at the origin:

  • Using spherical coordinates, $\rho = q \delta(\mathbf{r})$:
  $$\nabla^2 \phi = -\frac{q}{\epsilon_0} \delta(\mathbf{r}) \Rightarrow \phi(r) = \frac{q}{4\pi\epsilon_0 r} \quad (r \neq 0)$$

Laplace's Equation

• Statement: A special case of Poisson's equation when there are no sources:

$$\nabla^2 \phi = 0$$

• Physical Interpretation: The potential $\phi$ is harmonic, indicating no charge density in the region.

• Example: In spherical coordinates, the solution outside a charged sphere:

$$\phi(r) = \frac{C}{r} \quad (r > R)$$

6. Continuity Equation

Statement: Represents the conservation of charge:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0$$

• Physical Interpretation: The rate of change of charge density $\rho$ in a volume plus the divergence of current density $\mathbf{J}$ must sum to zero.

• Example: If $\rho = \rho_0 e^{-\lambda t}$:

  • The continuity equation implies:

$$\frac{\partial \rho}{\partial t} = -\lambda \rho_0 e^{-\lambda t} \Rightarrow \nabla \cdot \mathbf{J} = -\lambda \rho$$

• This shows how charge flows out of the volume.

7. Maxwell's Equations

Maxwell's equations unify electricity and magnetism and can be expressed in both differential and integral forms.

1. Gauss's Law:

   • Differential Form:
   $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
   • Integral Form:
   $$\iint_S \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\epsilon_0}$$
   • Example: For a uniform charge distribution $Q$ over a sphere of radius $R$:
   $$\iint_S \mathbf{E} \cdot d\mathbf{S} = E(4\pi R^2) = \frac{Q}{\epsilon_0} \Rightarrow E = \frac{Q}{4\pi\epsilon_0 R^2}$$

2. Gauss's Law for Magnetism:

   • Differential Form:
   $$\nabla \cdot \mathbf{B} = 0$$
   • Integral Form:
   $$\iint_S \mathbf{B} \cdot d\mathbf{S} = 0$$
   • Interpretation: This indicates that there are no magnetic monopoles; magnetic field lines are closed loops.

3. Faraday's Law of Induction:

   • Differential Form:
   $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
   • Integral Form:
   $$\oint_C \mathbf{E} \cdot d\mathbf{r} = -\frac{d\Phi_B}{dt}$$
   • Example: If a magnetic field through a loop changes with time, it induces an electric field around the loop, demonstrating how a changing magnetic field produces electric currents.

4. Ampère's Law (with Maxwell's correction):

   • Differential Form:
   $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
   • Integral Form:
   $$\oint_C \mathbf{B} \cdot d\mathbf{r} = \mu_0 I_{\text{enc}} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$
   • Example: For a long straight current-carrying wire, the magnetic field $B$ at a distance $r$ is given by:
   $$B = \frac{\mu_0 I}{2\pi r}$$

8. Physical Significance of Maxwell's Equations

• Unification of Electricity and Magnetism: Maxwell's equations show that electric fields and magnetic fields are interconnected. Changing one can produce the other.
• Prediction of Electromagnetic Waves: These equations predict that oscillating electric and magnetic fields can propagate through space as electromagnetic waves (e.g., light).
• Foundation for Modern Physics: They serve as the foundation for many areas in physics and engineering, including telecommunications, optics, and electromagnetic theory.

Examples:

1. Gradient ($\nabla f$)

Equation:

$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$

Example: Let $f(x, y, z) = x^2 + y^2 + z^2$.

Calculation:

$$\nabla f = \left( 2x, 2y, 2z \right)$$

Explanation: The gradient vector $\nabla f$ gives the direction and rate of steepest ascent of the scalar function $f$. At the point (1, 2, 3):

$$\nabla f(1, 2, 3) = (2, 4, 6)$$

This means that if you move in the direction of the vector $(2, 4, 6)$, you will experience the greatest increase in the function $f$.

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2. Divergence ($\nabla \cdot \mathbf{A}$)

Equation:

$$\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

Example: Let $\mathbf{A} = (x^2, y^2, z^2)$.

Calculation:

$$\nabla \cdot \mathbf{A} = \frac{\partial (x^2)}{\partial x} + \frac{\partial (y^2)}{\partial y} + \frac{\partial (z^2)}{\partial z} = 2x + 2y + 2z$$

Explanation: The divergence $\nabla \cdot \mathbf{A}$ tells us how much the vector field is "spreading out" from a point. If $(x, y, z) = (1, 1, 1)$:

$$\nabla \cdot \mathbf{A} = 2(1) + 2(1) + 2(1) = 6$$

This indicates that at this point, the field is expanding.

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3. Curl ($\nabla \times \mathbf{A}$)

Equation:

$$\nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}, \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}, \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right)$$

Example: Let $\mathbf{A} = (0, 0, z)$.

Calculation:

$$\nabla \times \mathbf{A} = \left( 0 - 0, 0 - 0, 0 - 0 \right) = (0, 0, 0)$$

Explanation: The curl of $\mathbf{A}$ is zero, indicating there is no rotational tendency at any point in this field. The field is irrotational, which means it does not induce rotation.

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4. Gauss's Law

Equation:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

Example: Consider a point charge $Q$ located at the origin.

Calculation: For a spherical surface of radius $R$ centered at the charge:

$$E = \frac{Q}{4\pi \epsilon_0 R^2}$$

The total electric flux through the sphere is:

$$\Phi_E = E \cdot A = \frac{Q}{4\pi \epsilon_0 R^2} \cdot 4\pi R^2 = \frac{Q}{\epsilon_0}$$

Explanation: Gauss's Law states that the electric flux through a closed surface is proportional to the enclosed charge. In this case, we see that all the electric field lines from the charge pass through the surface, confirming the relationship.

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5. Gauss's Law for Magnetism

Equation:

$$\nabla \cdot \mathbf{B} = 0$$

Example: Consider a magnetic field around a straight current-carrying wire.

Calculation: If you use a cylindrical surface around the wire, the magnetic field $B$ is tangential and has no divergence:

$$\nabla \cdot \mathbf{B} = 0$$

Explanation: This equation implies that there are no magnetic monopoles; magnetic field lines form closed loops. The absence of divergence means that all magnetic field lines that enter a volume must also exit it.

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6. Faraday's Law of Induction

Equation:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

Example: Consider a changing magnetic field through a loop.

Calculation: If the magnetic field $B(t) = B_0 \sin(\omega t)$ is changing, then:

$$\frac{\partial \mathbf{B}}{\partial t} = B_0 \omega \cos(\omega t)$$

The induced electric field around the loop is given by:

$$\oint_C \mathbf{E} \cdot d\mathbf{r} = -\frac{d\Phi_B}{dt} = -\pi R^2 \frac{dB}{dt}$$

Explanation: This law states that a changing magnetic field induces an electric field. The negative sign indicates that the induced electric field opposes the change in magnetic flux (Lenz's Law).

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7. Ampère's Law (with Maxwell's correction)

Equation:

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Example: For a long straight wire carrying a current $I$.

Calculation: The magnetic field at a distance $r$ from the wire:

$$B = \frac{\mu_0 I}{2\pi r}$$

The curl of $\mathbf{B}$ yields:

$$\nabla \times \mathbf{B} = \frac{\mu_0 I}{2\pi r}$$

Explanation: Ampère's Law states that the curl of the magnetic field is related to the current density $\mathbf{J}$ plus a term that accounts for changing electric fields. This reflects the unification of electric currents and changing electric fields in generating magnetic fields.

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8. Poisson's Equation

Equation:

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0}$$

Example: For a point charge $Q$ at the origin.

Calculation: In spherical coordinates, outside the charge:

$$\nabla^2 \phi = 0 \quad (r > 0)$$

Inside:

$$\nabla^2 \phi = -\frac{Q}{\epsilon_0} \delta(r)$$

Explanation: Poisson's Equation relates the Laplacian of the electric potential $\phi$ to the charge density $\rho$. Outside the charge, the potential behaves like $\phi(r) = \frac{Q}{4\pi \epsilon_0 r}$, indicating how the potential decreases with distance from the charge.

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9. Laplace's Equation

Equation:

$$\nabla^2 \phi = 0$$

Example: In a region without charges (like outside a charged sphere).

Calculation: For a spherically symmetric potential:

$$\phi(r) = \frac{C}{r}$$

where $C$ is a constant determined by boundary conditions.

Explanation: Laplace's Equation describes potential in regions without charge. The solution shows that potential behaves inversely with distance, indicative of how the potential fields spread out in space.