Unit1 - Subjective Questions

Scalar Field:

• A region in space where a scalar quantity is defined at every point is called a scalar field.
• It is specified by magnitude only.
• Examples: Temperature distribution in a rod, Electric potential (), Density of a body.
• Mathematically represented as .

Vector Field:

• A region in space where a vector quantity is defined at every point is called a vector field.
• It is specified by both magnitude and direction.
• Examples: Electric field intensity (), Magnetic field intensity (), Velocity of fluid flow.
• Mathematically represented as .

Definition:
The gradient of a scalar field is a vector field that represents the maximum rate of change of the scalar field in space.

Physical Significance:

1. Direction: $
   abla \phi\phi$.
2. Magnitude: The magnitude $|
   abla \phi|\phi$ along that direction.
3. Normal Vector: The gradient vector at any point is always normal (perpendicular) to the level surface (equipotential surface) passing through that point.

Physical Significance of Divergence:

• Divergence of a vector field at a point represents the net outward flux of the vector field per unit volume emanating from that point.
• Mathematically:
• Positive Divergence: Acts as a source (fluid expanding).
• Negative Divergence: Acts as a sink (fluid compressing).
• Zero Divergence: No loss or gain of fluid.

Solenoidal Vector:
If the divergence of a vector field is zero ($
abla \cdot \vec{A} = 0 \vec{B}$.

Concept and Significance:

• The Curl of a vector field , denoted by $
  abla \times \vec{A}$, represents the maximum circulation of the vector field per unit area.
• It indicates the tendency of the field to rotate or swirl around a point.
• The direction of the curl gives the axis of rotation.

Irrotational Field:

• If the curl of a vector field is zero ($
  abla \times \vec{A} = 0 $) everywhere, the field is called Irrotational (or conservative).
• Example: Electrostatic field .

Statement:
Gauss's Divergence Theorem states that the surface integral of the normal component of a vector function taken over a closed surface is equal to the volume integral of the divergence of that vector function taken over the volume enclosed by the surface .

Mathematical Representation:

Significance:
It relates a surface integral to a volume integral, facilitating the conversion between the two.

Statement:
Stokes' Theorem states that the line integral of a vector function around a closed path is equal to the surface integral of the curl of taken over the open surface bounded by the closed path .

Mathematical Representation:

Significance:
It relates a line integral to a surface integral, allowing conversion between the circulation of a field and the flux of its curl.

1. Consider a volume enclosed by a surface . The total current flowing out of the volume is given by the surface integral of current density :
   
2. According to the principle of conservation of charge, the current flowing out must equal the rate of decrease of charge inside the volume:
   
3. Total charge is the volume integral of charge density :
   
4. Therefore:
   
5. Applying Gauss's Divergence Theorem to the LHS:
   
6. Since this holds for any arbitrary volume, the integrands must be equal:
   
   
   This is the continuity equation.

Original Ampere's Law:

Identifying the Inconsistency:

1. Take the divergence of both sides:
   
2. Vector identity states that the divergence of the curl of any vector is always zero:
   
3. Therefore, it implies:
   
4. However, the equation of continuity states:
   
5. Conclusion: Ampere's original law implies $
   abla \cdot \vec{J} = 0\rho$ changes with time), the law is mathematically inconsistent.

Concept:

• Displacement current is not a flow of physical charge (electrons) but arises due to the time-varying electric field.
• In a capacitor circuit, conduction current flows through the wires, but no charge flows across the gap between plates. However, the circuit is complete.
• Maxwell proposed that the changing electric field () between the plates produces a current equivalent, called Displacement Current ().

Mathematical Formula:
Displacement current density is given by the rate of change of electric displacement vector :

1. Maxwell added a term to Ampere's law to fix the inconsistency:
   
2. Take divergence on both sides:
   
3. LHS is zero (div of curl):
   
4. From continuity equation, $
   abla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$. Thus:
   
5. From Gauss's Law for Electrostatics, $
   abla \cdot \vec{D} = \rho$. Differentiating with respect to time:
   
6. Comparing the expressions for :
   
7. Final Equation:
   

Conduction Current ():

• Cause: Due to the actual flow of free electrons/charge carriers.
• Medium: Flows through conductors.
• Law: Obeys Ohm's Law ().
• Condition: Can exist in steady (DC) and time-varying conditions.

Displacement Current ():

• Cause: Due to the time-varying electric field (change in electric flux).
• Medium: Flows through dielectrics or free space.
• Law: Follows .
• Condition: Exists only when the electric field is changing with time.

1. Gauss’s Law for Electrostatics:
   
2. Gauss’s Law for Magnetism:
   
3. Faraday’s Law of Electromagnetic Induction:
   
4. Modified Ampere’s Circuital Law:
   

(Where is electric displacement, is charge density, is magnetic flux density, is electric field, is magnetic field intensity, and is current density.)

1. Gauss’s Law for Electrostatics:
   
2. Gauss’s Law for Magnetism:
   
3. Faraday’s Law:
   
4. Modified Ampere’s Law:
   

1. Statement: The total electric flux passing through a closed surface is equal to the total charge enclosed by that surface.
   
2. We know that total charge is the volume integral of charge density :
   
3. Therefore:
   
4. Apply Gauss's Divergence Theorem to the LHS to convert surface integral to volume integral:
   
5. Comparing the two volume integrals:
   
6. Since the volume is arbitrary:
   

Physical Significance:

1. Non-existence of Monopoles: The equation states that the divergence of the magnetic flux density is always zero. This implies that isolated magnetic poles (magnetic monopoles) do not exist.
2. Closed Loops: Magnetic field lines are always continuous closed loops; they do not have a starting point (source) or ending point (sink) like electric field lines.
3. Flux Balance: The total magnetic flux entering a closed volume is exactly equal to the total magnetic flux leaving it.

1. From Faraday's Law of electromagnetic induction, the induced EMF is equal to the negative rate of change of magnetic flux:
   
2. EMF is defined as the line integral of the electric field :
   
3. Magnetic flux is the surface integral of :
   
4. Substitute these into Faraday's Law:
   
5. Apply Stokes' Theorem to the LHS (convert line integral to surface integral):
   
6. Comparing the surface integrals:
   
7. Since the surface is arbitrary:
   

1. $
   abla \cdot \vec{D} = \rho$ (Gauss's Law for Electrostatics):
   Electric charge is the source of the electric field. Field lines diverge from positive charges and converge on negative charges.
2. $
   abla \cdot \vec{B} = 0$ (Gauss's Law for Magnetism):
   Magnetic monopoles do not exist. Magnetic field lines are continuous closed loops. Net magnetic flux through a closed surface is zero.
3. $
   abla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Faraday's Law):
   A time-varying magnetic field generates an electric field. The electric field is non-conservative in this case (curl is non-zero).
4. $
   abla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$ (Modified Ampere's Law):
   Magnetic fields are generated by conduction currents (moving charges) and displacement currents (changing electric fields).

1. Start with Maxwell's first equation (Gauss's Law):
   
2. For a linear, isotropic, homogeneous medium, . Thus:
   
3. In electrostatics, the electric field is the negative gradient of the scalar potential :
   
4. Substitute this into step 2:
   
   
   
5. Poisson's Equation:
   
6. Laplace's Equation:
   If the region is charge-free (), Poisson's equation reduces to:
   

Electrostatic Field:

• Generated by static charges.
• $
  abla imes \vec{E} = 0$.
• Work done in moving a charge in a closed path is zero ().
• Hence, it is a conservative field and can be represented as the gradient of a scalar potential.

Induced Electric Field (Time-varying):

• Generated by time-varying magnetic fields (Faraday's Law).
• $
  abla imes \vec{E} = -rac{\partial \vec{B}}{\partial t}
  eq 0$.
• Work done in a closed path is not zero; it equals the EMF generated.
• Hence, it is a non-conservative field.

1. Start with the Equation of Continuity:
   
2. Steady Current Definition: A steady current implies that charge does not accumulate at any point in space over time. The flow is constant.
3. Therefore, the charge density is constant with respect to time:
   
4. Substituting this into the continuity equation:
   
   
   
5. This implies that for steady currents, the current density vector is solenoidal.