Phy 213: General Physics III

1. Phy 213: General Physics III Chapter 29: Magnetic Fields to Currents Lecture Notes

2. . P f i The Biot-Savart Law • The magnetic field analog to Coulomb’s Law is the Biot-Savart Law, which describes the magnetic field (dB) produced by a segment of current-carrying wire (ds) at a distance, r: • Integrating the Biot-Savart Law over the whole wire will yield the total magnetic field, B:

3. y . P x z f i Calculating B Field due to a straight wire • Consider a long straight wire (length = L) with current, i. The magnetic field (dB) at position P, directly above the center of the wire, due to a segment of the wire is given by the Biot-Savart Law: • Integrating over the length of the wire:

4. P Magnetic field for a straight wire • In general, the magnitude of the magnetic field due to a straight wire, at a distance r, is given by: • The direction of dB and the net B field is always perpendicular to the plane of the wire and point P: • The B field is said to “curl” around the current in the wire (as per RHR) • When the length of the wire (L) is much greater than the separation distance (r), the magnitude of the B-field reduces to:

5. Magnetic Field of a Circular Arc • The B field at the center of curvature of a circular arc can be determined by applying the Biot-Savart Law to a segment of the arc: • Integrating over the entire arc: where is the unit vector normal to the plane of the arc. Note: For a circular loop, f=2p: i R P .

6. i1 d i2 i1 d i2 Force Between Parallel Currents • When 2 parallel wires (length=L) both carry current, their respective B-field exerts a magnetic force on the other Attractive: Repulsive:

7. R i Ampere’s Law • The magnetic analog to Gauss’ Law is Ampere’s Law, which can be used in certain circumstances to calculate a B-field more simply than the Biot-Savart Law. Definition: for a closed path around an enclosed current Example: The B field outside a straight, infinitely long current carrying wire. • Define a circular loop (called an “Amperean Loop”) centered on the wire. Since the path is equidistant at all points to the wire, B will have constant magnitude along the loop. • Apply Ampere’s Law: • Solve for B:

8. Interpreting Ampere’s Law There is a more fundamental way of interpreting Ampere’s Law: • We begin w/ Ampere’s Law: • can be interpreted using Gauss’ Law: • Thus, Ampere’s Law becomes: Conclusion: i. A time varying E-field produces a corresponding B- field ii. The presence of a B-field implies the presence of a corresponding, time varying E-field

9. i N l Magnetic Fields in Coils (& solenoids) • Ampere’s Law is an effective tool for calculating the B-field inside a straight coil or solenoid, with N uniform turns: • Choosing a rectangular Amperean Loop of length, l. The B-field components around the path segments outside the solenoid equal zero, so Ampere’s Law reduces to: • Solving for the B-field: where n = N/l, the turn density for the solenoid. • The direction of the B-field is defined by the RHR

10. R i N Field of a Magnetic Dipole • A current carrying loop or coil is an effective magnetic dipole (analogous to a straight bar magnet), where the B-field is parallel to the area vector of the coil • The B-field at point P along z-axis, can be analyzed using the Biot-Savart Law: When z>>R, the B-field equation simplifies to: . P {the B-field for a magnetic dipole along Z-axis}