Class 12 Physics Moving Charges Magnetism Ampere Circuital Law

Ampere’s Circuital Law:

  • Ampere’s circuital law states that line integral of magnetic field forming a closed loop around the current(i) carrying wire, in the plane normal to the current, is equal to the μo times the net current passing through the close loop.

Here μo = permeability of free space = 4π×10-15N/A2

  • This law is based on the assumption that the closed loop consists of small elemental parts of length dl, and the total magnetic field of the closed loop will be the integral of magnetic field and the length of these elements This closed loop is called Amperian loop
  • Further, this integral will be equal to the multiplication of net current passing through this closed loop and the permeability of free space(μoi)

Proof-1(Regular coil):

To prove:        B.dl = μoi

Starting from the left hand side, we can see in the diagram that angle between the element dl and magnetic field B is

 We know that magnetic field due to a long current carrying wire is:

B = μoi/(2πr)

Also, the integral of element will form the whole circle of circumference (2πr):

∫ dl = 2πr

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

B∫ dl = μoi/(2πr) × 2πr = μoi

B.dl = μoi

 

Proof-2(Irregular coil):

To prove:        B.dl = μoi

Starting from the left hand side:

B.dl1 = μoi/(2πr1) × dl1

We know that:  1 = dl1/r1

μoi/(2πr1) × dl1 =μoi/(2π)1 = μoi

B.dl = μoi

Note:

  • The above two derivations proves that magnetic field at a point doesn’t depend on the shape of the Amperian loop.
  • Magnetic field is same at every point in the Amperian loop (magnetic field possesses cylindrical symmetry)
  • Direction of magnetic field at any point on the Amperian loop is tangential to the circle formed at that point with wire passing through the center, and the direction could be calculated by right hand thumb rule where, on holding the current carrying wire such that the extended thumb shows the direction of current in the wire, then the curling of rest of the 4 fingers represent the direction of rotation of magnetic field.

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