Biot-Savart Law

1. Biot-Savart Law describes the field contributions of tiny segments of current at a position external to the wire in which the current is flowing i Note: r-hat is a unit vector sinq comes from the cross product  angle between wire segment dl and position vector r

2. Field of a long wire i R x Note: angle here is direction of position vector relative to the horizontal, not angle between dx and r so we use cosine not sine. r dx Recall secant=1/cosine therefore

3. Force between Two Parallel Wires Since a current creates a b-field in the surrounding region, another current in the region will feel a force due to this field. If the current flows are parallel, what direction is the field created by i1 near i2 ? Into paper What direction is the force on a segment of i2 ? Up, toward the other wire. Parallel currents attract  check with curled RHR

4. We can rewrite our result for the long wire in a more general way. Recall that the field lines around the wire are concentric circles. ds B

5. Ampere’s Law Red is into paper

6. Ampere’s law will be useful in cases of high symmetry in solving for the magnetic field created by a current distribution. Just as in using Gauss’ law, we introduce a geometrical construction, in this case called an “Amperean loop” around which we compute the integral. This loop is chosen so that B will factor out of the integral and can then be solved for. The left side of this equation is determined by the symmetry of the problem, and will be the same for all problems with the same symmetry.

7. Field inside a wire carrying uniform current distribution a r B r a

8. Field inside a uniform (j) sheet of current L x t Only contributions along L and its partner

9. Solenoid A solenoid is a single wire wrapped in multiple loops or “windings”. It is characterized by the number of windings per meter, n, the current, i, its length l and its cross sectional area A. In the limit of an infinitely long solenoid, the field outside the coil vanishes. We can apply Ampere’s law to find B inside the coil. s o o o o o o Only the leg inside coil, parallel to B will contribute. x x x x x x Note that B is uniform

10. Toroid A toroid is basically a finite solenoid bent into the shape of a donut. It is characterized by an inner radius a, an outer radius b, a current i, and a total number of windings N. o o b x o x x o By considering these Amperean paths, one can argue that the field is identically zero outside the toroid. x a Consider the circular path of radius r inside the toroid.