# Ampere’s Circuital Law

What is Ampere’s Circuital Law? Well, it is a current distribution which helps us to calculate the magnetic field. And yes, the Biot-Savart law does the same but Ampere’s law uses the case high symmetry. We will first understand the ampere’s circuital law, followed by its proof. So let us begin!

## Ampere’s Circuital Law

What is stated by Ampere’s Circuital Law? The formula for this is a closed loop integral. The integral of magnetic field density (B) along an imaginary closed path is equal to the product of current enclosed by the path and permeability of the medium. Line integral to the magnetic field of the coil = μtimes the current passing through it. It is mathematically expressed as

∫ B.dl =  μI

Here μo = permeability of free space = 4 π × 10-15 N/ Aand ∫ B.dl = line integral of B around a closed path.

## Proof of Ampere’s Circuital Law

### Case 1: Regular Coil

Consider a regular coil, carrying some current I. Let us assume a small element dl on the loop.

∫B dl =  ∫B dl cos θ

Here, θ is the small angle with the magnetic field. The magnetic field will be around the conductor so we can assume,

θ = 0°

We know that, due to a long current-carrying wire, the magnitude of the magnetic field at point P at a perpendicular distance ‘r’ from the conductor is given by,

B =  $$\frac{μ_0i}{2πr}$$

The magnetic field doesn’t vary at a distance r due to symmetry. The integral of an element will form the whole circle of the circumference (2πr):

∫ dl = 2πr

Put the value of B and ∫ dl in the equation, we get:

B∫ dl = $$\frac{μ_0i}{2πr}$$ × 2π r = μoi

therefore, ∫ B.dl = μoi

### Case 2: Irregular Coil

Irregular coil means a coil of any arbitrary shape. Here the radius will not remain constant as it is not a regular coil.

∫ B.dl1 = ∫ $$\frac{μ_0i}{2πr}$$ × dl1

As we  know  :  1 = $$\frac{dl_1}{r_1}$$

∴∫$$\frac{μ_0i}{2πr}$$ × dl1 = $$\frac{μ_0i}{2π}$$∫dθ1 = μoi

∫ B.dl = μoi

So whether the coil is a regular coil or an irregular coil, the ampere’s circuital law holds true for all.

## Amperian Loop

Ampere’s circuit law uses the Amperian loop to find the magnetic field in a region. The Amperian loop is one such that at each point of the loop, either:

• B is tangential to the loop and is a non zero constant
•  or B is normal to the loop, or
• B vanishes

where  B is the induced magnetic field.

## Solved Examples for You

Q1. Mark the incorrect option.

1. Amperes law states that the flux B through any closed surface is μtimes the current passing through the area bounded by a closed surface.
2. Gauss’s law of magnetic field serves the same purpose as the Gauss’s law for the electric field.
3. Gauss’s law of magnetic field states that the flux of B in any closed surface is equal to zero, whether there are or bot any currents within the surface.
4. All of the above.

Solution: A. Ampere law states that for any close looped path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. Option A is correct.

Q2. A student gets confused if two parallel wires carrying current in the same direction attract or repel. Which rules will he need to reach the right conclusion?

1. Right-Hand Thumb Rule
2. Fleming Heft Hand Rule
3. Both A and B
4. None

Solution: C. Consider two parallel wires carrying current in the same direction. When right-hand thumb rule and Fleming left-hand rule is applied, it is observed that the force in the direction of the first wire i.e second wire is attracted to the second wire. Similarly, the second wire is also attracted to the first wire. Hence they attract.

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