Lecture 15 1 Sources of Magnetic Fields Moving

Lecture 15 -1 Sources of Magnetic Fields • Moving point charge: also Permeability constant • Bits of current: I Biot-Savart Law The magnetic field “circulates” around the wire. http: //falstad. com/vector 3 dm/

Lecture 15 -2 Magnetic field due to a current loop Principle of superposition: At the center, On axis generally,

Lecture 15 -3 Magnetic Field At Center of Circular Arc Current I runs clockwise in the closed loop wires below: What is B at center?

Lecture 15 -4 Circular Loop Current as a Magnetic Dipole if |x|>>R In fact, On electric dipole axis

Lecture 15 -5 READING QUIZ 1 Which of the following statements about static magnetic fields is correct? A| Magnetic fields lines end and begin in magnetic charges called monopoles. B| The magnetic flux through a closed surface can have a non-zero value. C| Magnetic field lines are closed loops which link (surround) currents. D| Ampere’s law contradicts the Biot-Savart law.

Lecture 15 -6 Gauss’s Law for Magnetism sources Gauss’s Law for Magnetism No sources

Lecture 15 -7 Magnetic field of a solenoid L • A constant magnetic field could be produced by an infinite sheet of current. In practice, however, it is easier and more convenient to use a solenoid. • A solenoid is defined by a current I flowing through a wire that is wrapped n turns per unit length on a cylinder of radius R and length L. Stretched-out solenoid R

Lecture 15 -8 Magnetic field of a solenoid (continued) Contribution to B at origin from length dx one turn # turns in length dx (or for L>>R) (half at ends)

Lecture 15 -9 Solenoid’s B field synopsis • Long solenoid (R<<L): B inside solenoid // to axis B outside solenoid nearly zero (not very close to the ends or wires) Solenoid’s B field Bar magnet’s B field

Lecture 15 -10 PHYS 241 – Warm-Up Quiz 2 A circular current loop lies in the plane perpendicular to this sheet with its axis along the x-direction, and produces magnetic field B as shown. What is the direction of the current at the top end of this loop? a. Out of the sheet b. Into the sheet c. Along +x axis d. Along -x axis e. Current is zero. x

Lecture 15 -11 Magnetic field of a Straight Current y I Infinite straight current

Lecture 15 -12 Definition of charge 1 C (1 A for 1 Sec) Two Parallel Currents The ampere is defined to be the constant parallel currents that will produce the force between them of 2 x 10 -7 N per meter when 1 m apart.

Lecture 15 -13 B Two Perpendicular Currents FB 2, 1 FB 1, 2 I 2 B I 1 I 3 FB 1, 3

Lecture 15 -14 DOCCAM 2

Lecture 15 -15 Force on a Moving Charge near a Current Magnetic Force (a) I B (b) q>0 (a) (c) (b) B B (c) zero

Lecture 15 -16 Ampere’s Law in Magnetostatics Biot-Savart’s Law can be used to derive another relation: Ampere’s Law The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, • Choosing a direction of integration. • A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration.

Lecture 15 -17 Example: Magnetic field of a long wire outside the wire inside the wire

Lecture 15 -18 Example: A Non-Uniform Current Distribution Long, hollow cylindrical current of current density: Insider the cylinder, the total current encircled by the Amperian loop is

Lecture 15 -19 Ampere’s Law applied to a solenoid • Long solenoid (a<<L): B inside solenoid // to axis B outside solenoid nearly zero (not very close to the ends or wires) • Ampere’s Law: n windings per unit length B = μO n h I n = N/l

Lecture 15 -20 Limitations of Ampere’s Law E q Not enough symmetry -q Ampere’s Law needs correction!

Lecture 15 -21 PHYS 241 – 10: 30 Quiz 3 A circular current loop lies on the xy-plane as shown, where the current is clockwise as seen from the positive z-axis. What is the direction of the B field at point A? z a. Along +x axis b. Along +y axis c. Along +z axis d. Along -z axis e. Along -x axis y I x A

Lecture 15 -22 PHYS 241 – 11: 30 Quiz 3 A circular current loop lies on the xy-plane as shown, where the current is clockwise as seen from the positive zaxis. What is the direction of the B field at point A? z a. Along +x axis b. Along +y axis c. Along +z axis d. Along -z axis e. Along - x axis y I A x

Lecture 15 -23 PHYS 241 - Quiz C A circular current loop lies on the xy-plane as shown, where the current is clockwise as seen from the positive zaxis. What is the direction of the B field at point A? z a. Along +x axis b. Along +y axis c. Along +z axis d. Along z axis e. Along x axis y A I x