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

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: i. P 

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: Calculating B Field due to a straight wire i. P  x y z

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: 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:

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,  =2  : R P. i

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: i1i1 i2i2 d i1i1 i2i2 d

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. 1.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. 2.Apply Ampere’s Law: 3.Solve for B: i R

Interpreting Ampere’s Law There is a more fundamental way of interpreting Ampere’s Law: 1.We begin w/ Ampere’s Law: 2. can be interpreted using Gauss’ Law: 3.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

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: 1.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: 2.Solving for the B-field: where n = N/ l, the turn density for the solenoid. 3.The direction of the B-field is defined by the RHR i N l

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: {the B-field for a magnetic dipole along Z-axis} i N R. P

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