Sources of Magnetic Fields Moving point charge: Bits of current: I also Permeability constant Biot-Savart Law The magnetic field “circulates” around the wire. http://falstad.com/vector3dm/

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

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

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

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.

Gauss’s Law for Magnetism sources Gauss’s Law Gauss’s Law for Magnetism No sources

Magnetic field of a solenoid 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. R 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

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

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

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

Magnetic field of a Straight Current y I Infinite straight current

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. Definition of charge 1 C (1 A for 1 Sec)

Two Perpendicular Currents FB 1,2 I1 FB2,1 B I2 I3 FB 1,3

DOCCAM 2

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

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.

Example: Magnetic field of a long wire outside the wire inside the wire

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

Ampere’s Law applied to a solenoid // to axis Long solenoid (a<<L): B inside solenoid 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

Limitations of Ampere’s Law q -q Not enough symmetry Ampere’s Law needs correction!

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

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 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

PHYS241 - Quiz C 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 A x