The magnetic field of a current-carrying conductor can be simply given by the current flowing through the conductor μₒ times. The stationary charge produces the magnetic field proportional to the magnitude of the charge but the moving charge produces a magnetic field proportional to the current.

The main drawback to Ampere’s Law is that differential equation, which means it requires basic calculus to apply. When you employ Ampere’s Law, you look at the circumstance you’re in, plug some values, and solve the integral. This will provide you with an equation that is appropriate for the condition.

## André-Marie Ampère

French physicist, a mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as “electrodynamics”. He is also the inventor of numerous applications, such as the solenoid and the electrical telegraph.

## Ampere’s Circuital Law

The ampere law states the line integral of the magnetic field around any closed path is equal to μ0 the current passing through it in the closed path.

This law assumes that the closed-loop is made up of small elemental portions with a length of dl and that the total magnetic field of the closed-loop is equal to the integral of the magnetic field and the length of these elements. Ampere loop is the name given to this closed loop.

This integral will also be equal to the product of the net current going through the closed-loop and the permeability of free space.

B.dl=0I

Where

B = Magnetic field

μₒ = 4π x 10⁻⁷ N/A² = permeability of free space

I = Current

##### The following result can be derived from the above equations

1. The preceding equation shows that the magnetic field at a given position is independent of the Ampere loop’s geometry.

2. Every point in the Ampere loop has the same magnetic field (magnetic field possesses cylindrical symmetry)

The direction of the magnetic field at any point on the Ampere loop is tangential to the circle formed at that point with a wire passing through the center, and it can be calculated using the right-hand thumb rule, which involves holding the current-carrying wire so that the extended thumb shows the direction of current in the wire, and then curling the rest of the four fingers to represent the direction of magnetic field rotation. The Right-hand rule is also known as Flemings’ right-hand rule.

## Magnetic Field

Suppose we have to find the magnetic field at a distance of d from the current-carrying wire then we have to draw an imaginary circle around the current-carrying conductor of radius d.

So the magnetic field is given by

H.dl=2 dH

Where

H=I cl /2r

I enclosed is the current flowing through the wire which is closed in the circle.

## Right-hand thumb rule

For the direction of the magnetic field, we have to curl the fingers around the current-carrying conductor such that the direction of the thumb shows the direction of the current and so the curled fingers shows the direction of the magnetic field. The right-hand rule is a physics rule for identifying the direction of three-dimensional axes or parameters. The right-hand rule, invented by British physicist John Ambrose Fleming in the 19th century for electromagnetism applications, is most commonly used to determine the direction of a third parameter when the other two are known (magnetic field, current, magnetic force).

## Maxwell’s equation of Ampere’s law

Ampere was a scientist who experimented with stresses on electric current-carrying wires. In the 1820s, he was conducting these tests about the same time that Faraday was developing Faraday’s Law. Ampere and Faraday had no idea that their work would be merged nearly 40 years later by Maxwell himself.

Ampere discovered the first term on the right-hand side of the equation. Around any closed contour line, it depicts the relationship between a current I and the circulation of the magnetic field b.

∇ x H = D/t + J

Where

J = electric current

D = Flux Density

H = field