# Ampere's Law

## Magnetics of Class 12

The integral of the quantity . daround loop is related to the current flowing through the surface bounded by the loop.

. d = μoI (3.22)

Note that there are often contribution to from currents not enclosed by the loop. This law may be used to find if the geometry of the currents allow the choice of a path for which the integral may be easily evaluated.

### 1. Infinitely long straight wire,

Consider a circular path of radius r as shown in the fig. (3.24)

. d = ∫Bdl cos0o = B ∫ dl = B (2πr)

Since the current enclosed by the circular path is I, therefore, using Ampere’s law

B (2πr) = μoI

or B = μ0I/2πr

### 2. Infinite current sheet

Consider an infinite large sheet of current as shown in the figure (3.25) where j is the linear current density i.e. the current per unit length. Imagine a rectangular loop around the sheet as shown.

. d = μoI

ab. d +∫dc . d + ∫cd. d + ∫da. d = μoI

Since ⊥ d in the path b → c and d → a, therefore

bc. d = ∫da. d= 0

Moreover along the paths a → b and c → d, is parallel to path directions. Thus

ab. d + ∫cd. d = 2Bl

The current enclosed is I = j l

Thus, applying Ampere’s law

2B l = μo(j l)

or B = μ0j/2 (3.23)

### 3. Toroidal Solenoid

A toroid consisting of N turns is shown in the figure (3.26 a).

The magnetic field lines are closed rings as shown in fig. (3.26 b).

Applying Ampere’s law to a loop of radius r

B ∫dl = B (2πr) = μo (NI)

Then B = μ0NI/2πr a ≤ r ≤ b (3.24)

Note that the magnetic field varies as I / r within a toroid.