Maxwell's modified form of Ampere's circuital law is

Options

A. `ointvecB.vec(ds)=0`
B. `ointvecB.vec(dl)=mu_(0)I`
C. `ointvecB.vec(dl)=mu_(0)I+(1)/(epsilon_(o))(dq)/(dt)`
D. `ointvecB.vec(dl)=mu_(0)I+mu_(0)epsilon_(o)(d oint_(E))/(dt)`

Answer

To derive Maxwell's modified form of Ampere's circuital law, we need to understand the original Ampere's circuital law and the modifications introduced by Maxwell.

### Step-by-Step Solution:

1. **Understanding Ampere's Circuital Law**:
- The original Ampere's circuital law states that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is equal to the permeability of free space \( \mu_0 \) times the current \( I \) passing through the loop.
- Mathematically, it is expressed as:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \]

2. **Identifying the Limitation**:
- This law assumes that there is a steady current flowing through the wire. However, it does not account for situations where the electric field is changing, such as in the case of a capacitor where there is no conduction current, yet a magnetic field is still present.

3. **Introducing Displacement Current**:
- To address this limitation, Maxwell introduced the concept of displacement current. He recognized that a changing electric field can also produce a magnetic field, similar to how a conduction current does.
- The displacement current \( I_d \) is defined as:
\[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \]
- Where \( \Phi_E \) is the electric flux and \( \epsilon_0 \) is the permittivity of free space.

4. **Modified Form of Ampere's Circuital Law**:
- Incorporating the displacement current into the original Ampere's circuital law, we get:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I + I_d \right) \]
- This can be rewritten as:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I + \epsilon_0 \frac{d\Phi_E}{dt} \right) \]

5. **Final Expression**:
- Thus, Maxwell's modified form of Ampere's circuital law is:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I + \epsilon_0 \frac{d\Phi_E}{dt} \right) \]