Find the magnetic field on the axis of a current loop using the Biot-Savart law for magnetic field.
We compute the magnetic field of a ring of current on the symmetry axis of the ring. To find the magnetic field on the axis of a current loop, we apply the Biot-Savart law for magnetic field.
We start by highlighting a small current increment on the loop called dl (vector), which points in the direction of the current. We define a position vector pointing from the current increment to the observation point on the axis of the ring. Then we define r-hat, the unit vector in the direction of the position vector r. The cross product in the Biot-Savart law requires us to cross dl into r-hat, and this results in a magnetic field contribution at the observation point.
Next, we argue that only the horizontal component of the magnetic field will survive here, because every dl is paired with another dl on the other side of the ring, causing the off-axis contributions to magnetic field to cancel, so the net magnetic field of a ring of current must point along the symmetry axis of the ring. This requires us to use the cosine function to pick out the x component of the magnetic field, and we relate this to an angle in the original diagram by doing a bit of trig.
Now we can start setting up the Biot-Savart integral. We can compute the magnetic field integral for the scalar Bx (the magnitude of the horizontal component of the field, because vector magnitudes simply add together when every vector points in the same direction!) Finding the magnitude of the cross product, and applying the pythagorean theorem to compute the magnitude of the position vector and the cosine of theta, we arrive at an expression that is ready to integrate to find magnetic field.
The integral is trivial this time: everything is a constant except dl, and when we integrate around the ring, the integral is just equal to the circumference, 2pi*R. Simplifying our result, we get the field of a current loop using the Biot-Savart law for magnetic field: mu_0*IR^2/(2*(R^2+s^2)^3/2), where s is the distance to the center of current loop.
Taking the small s limit, we obtain the notable formula for the magnetic field at the center of a ring: mu_0I/2R.
#physics #magnetism #apphysics
