Ampere's Law

Ampere's Law

• Ampere's Law relates the magnetic field around a closed loop to the electric current passing through the loop.

• Full version (requires calculus):

  $\oint B \cdot dl = \mu_0 I$

• Simplified version (without path integrals):

  $\sum B{\parallel} \Delta l = \mu0 I$

  • $B_{\parallel}$ is the component of the magnetic field parallel to the length element $l$.

  • The summation is over a closed loop.

  • The right side represents the amount of current crossing the closed loop.

Application to a Long Straight Wire

• Consider a circle of radius $r$ around a long straight wire.

• The sum of length elements around the circle is $2 \pi r$.

• If the current intercepted is $I$, then:

$B (2 \pi r) = \mu0 I$ $B = \frac{\mu0 I}{2 \pi r}$

Application to a Solenoid

• Inside the solenoid, the magnetic field $B$ points upwards.

• Consider a rectangular path of length $l$ down the center of the solenoid.

• Applying Ampere's Law: $B l = \mu_0 N I$

  • $N$ is the number of wires going through the loop.

  • $I$ is the current.

• Therefore: $B = \mu0 \frac{N}{l} I = \mu0 n I$

  • Where $n$ is the number of turns per unit length.

Practical Application

• Solenoids can be created by winding a wire around a cylindrical form to create a magnetic field. This is a practical application of using Ampere's Law to enhance and control magnetic fields for various electromagnetic devices.