Ampere's Circuital Law and its meaning

Ampere's Circuital Law
The Ampere's Circuital Law without Maxwell's correction

In the 1820s, Ampere first identified that all magnetic effects are caused by the charged particles in motion, i.e., current. Just as the role played by the Gauss's law in electrostatics, in magnetostatics is the Ampere's circuital law plays the same role. Being one of Maxwell's equations, the Ampere's law describes a relationship between the current and the magnetic field it produces. Today, I am going to discuss in detail what does Ampere's circuital law really means.

The statement of this law is as follows:

The curl of the magnetic field intensity at a point is equal to the volume current density at that point. 

To understand this law, let us first try to understand the physical meaning of the Curl of a vector field.

The Curl of a Vector

The curl of a vector field measures the tendency for the vector field to swirl around. Imagine that the vector field represents the velocity vectors of water in a lake. If the vector field swirls around (rotates), then when we stick a paddlewheel into the water, it will tend to spin. The amount of the spin will depend on how we orient the paddle. Thus, we should expect the curl to be vector-valued.

Ampere's law ( DIFFERENTIAL FORM):

Imagine a magnetic field which is rotational or whose magnetic field lines formed curved paths. The curl of such a magnetic field at a point is equal to the product of permeability constant and volume current density at that point.

Integral form of Ampere's law:

Integral form of Ampere's circuital law indicates that the volume current density at any point in space is proportional to the spatial rate of change of the magnetic field. This volume current density is also perpendicular to the magnetic field at that point.

A simple way to understand this- If we sum up the magnetic field at all points of an imaginary loop drawn around a current-carrying conductor, it turns out to be proportional to the current enclosed by that imaginary loop. This sum of magnetic fields at all points is equal in magnitude to permeability in vacuum times the enclosed current.

Physical Meaning of Ampere's law:

Ampere's Circuital Law
The integral form of Ampere's law

If we just go back to our high school science, we recall that a current-carrying wire produces a magnetic field around it. The direction of this magnetic field can be obtained by using the right-hand thumb rule. If we align the thumb of our right hand in the direction of current flowing through a conductor, the curling of the fingers would indicate the direction of magnetic field lines.

So, we know that the magnetic field is in the form of closed loops around a current-carrying conductor. This looping is either clockwise or anti-clockwise. So far, so good! Now, what about the magnitude of this magnetic field that the current has generated around the wire through which it is flowing. Well, the easiest way to determine it is by using Ampere's law. (See how?)

Previous in Month of Equations: Mathematics Behind Heisenberg’s Uncertainty Principle.

Author's Message

The magnetic forces have surprisingly much smaller amplitudes than electric forces. Then how do we can easily notice the magnetic forces around a current-carrying wire? This is because of the reason that even though a huge amount of negative charges are flowing down the wire, an equal positive charge is already present in the wire. So, all the electric effects get masked and the magnetic field stands alone. Ampere's law talks about something that is so fundamental but still astonishing to understand. I hope this article gave you a brief insight into one of Maxwell's equations. Happy reading!

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