Ampere's circuital law can be derived from

Options

A. Ohm's law
B. Biot-Savart's law
C. Kirchhoff's law
D. Gauss's law

Answer

To derive Ampere's Circuital Law, we can start from the Bio-Savart Law. Here’s a step-by-step solution:

### Step 1: Understand the Bio-Savart Law
The Bio-Savart Law states that the magnetic field \( B \) at a point in space due to a small segment of current-carrying wire is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \, \sin \theta}{r^2} \]
where:
- \( dB \) is the infinitesimal magnetic field produced by the current segment,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the wire,
- \( dL \) is the length of the wire segment,
- \( \theta \) is the angle between the wire segment and the line connecting the segment to the point of observation,
- \( r \) is the distance from the wire segment to the point of observation.

### Step 2: Integrate the Bio-Savart Law
To find the total magnetic field \( B \) at a point due to a complete current loop, we integrate the Bio-Savart Law over the entire loop:
\[ B = \int dB = \int \frac{\mu_0}{4\pi} \frac{I \, dL \, \sin \theta}{r^2} \]

### Step 3: Apply Ampere's Circuital Law
Ampere's Circuital Law states that the line integral of the magnetic field \( B \) around a closed loop is equal to \( \mu_0 \) times the total current \( I_{\text{enc}} \) enclosed by the loop:
\[ \oint B \cdot dL = \mu_0 I_{\text{enc}} \]

### Step 4: Relate the Two Laws
The relationship between the two laws can be established by considering a closed loop where the current is flowing. The integration of \( B \cdot dL \) around the loop gives us the total magnetic field produced by the currents enclosed within that loop.

### Conclusion
Thus, we can conclude that Ampere's Circuital Law can be derived from the Bio-Savart Law, as it provides a way to calculate the magnetic field due to current-carrying conductors in a closed loop.

### Final Answer
Ampere's Circuital Law can be derived from the Bio-Savart Law.

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