Magnetic Field

The magnetic field inside a solenoid is given by: where is the number of turns per unit length, is the current, and is the permeability of free space. The energy density of a magnetic field is given by: where is the total energy stored in the magnetic field and is the volume. The energy stored in a magnetic field is given by: For a uniform magnetic field inside the solenoid, we have: Thus, the magnetic energy density of the magnetic field is .

The magnetic field at a point on the axis of a circular current-carrying loop can be derived using the Biot-Savart law: For a point on the axis, is the distance from the element of the loop to the point where the field is calculated. By integrating the contributions from all current elements on the loop, we get the expression for the magnetic field at a point on the axis of the loop: At the center of the loop (when ): Thus, the magnetic field at the center of the loop is , where is the radius of the loop.