Class 33. Amperes Law Path Integral of Magnetic Field Path Integral: - PowerPoint PPT Presentation

Class 33. Ampere’s Law

Path Integral of Magnetic Field Path Integral:     B d s ds B Do you remember:      (for stationary case) E d s ?

Gauss’s Law (Maxwell’s first equation) From Class 6 For any closed surface,          q or E d A q 0 E in 0 in Two types of problems that involve Gauss’s Law: 1. Give you left hand side (i.e. flux through a given surface), calculate the right hand side (i.e. charge enclosed by that surface). 2. Give you right hand side (i.e. a charge distribution) , calculate the left hand side (i.e. flux and the electric field).

Ampere’s Law (Maxwell’s third equation ‐ partial) I 1 I 2 For any closed loop,       B d s 0 I in      7 -1 4 10 TmA 0 Two types of problems that involve Ampere’s Law: 1. Give you left hand side (i.e. line integral of a given loop), calculate the right hand side (i.e. current enclosed by that loop). 2. Give you right hand side (i.e. current) , calculate the left hand side (i.e. the line integral and the magnetic field).

Calculating Magnetic Field Using Ampere’s Law 1. By symmetry argument, construct a loop so that     the path integral can be easily calculated. B d s In most cases when Ampere’s Law is applicable,      B d s BL where L is the length of the loop. 2. You can then apply Ampere’s Law and solve for B:    I          0 in B d s I BL I B 0 in 0 in L 3. Two common cases when Ampere’s Law can be used to calculate magnetic field: infinite long wire and infinite long solenoid / toroid.

Magnetic field due to a long wire Want to calculate the magnetic field B B at point P. P By symmetry argument, B is in the plane of the paper (infinite long r wire), has the same magnitude for I all points on the dotted circular loop (azimuthal symmetry), and tangent to the circular loop (so cos  =1).         B d s B 2 r            Ampere’s Law: B d s I B 2 r I 0 0  I   0 B  2 r

Magnetic Force Between Two Parallel Long Wires Magnetic field at point P due to I 1 :  B I  0 1 B  2 r P If another current I 2 parallel to I 1 is If another current I 2 parallel to I 1 is r passing through point P, it will passing through point P, it will I 1 experience a force because of the experience a force because of the field there. field there.

Magnetic Force Between Two Parallel Long Wires Magnetic field at point P due to I 1 :  I B  0 1 B  2 r P If another current I 2 parallel to I 1 is If another current I 2 parallel to I 1 is r F B passing through point P, it will passing through point P, it will I 1 experience a force because of the experience a force because of the field there. field there.         o F I L B F I BL sin 90 I BL B 2 B 2 2  I    0 1 F I L  B 2 2 r  F I I   B 0 1 2  L 2 r Force is attractive if the two currents are in the same direction, repulsive if the two currents are in opposite direction.