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.

Magnetization and “Bound” Current in Matter Note cancellations of internal currents from adjacent loops Strong externally applied field Bapp aligns the magnetic moments in matter. Magnetization Ampere: Aligned magnetic moments in magnetized matter arise due to microscopic current loops inside the material. A Bound current magnetic moment dµ due to Amperian current di current /length Equivalent to a solenoid of nI=M

MAGNETIC MATERIAL TYPES Magnetic materials are placed inside a solenoid with a magnetic field Bs. The following kinds of materials change the resultant B inside the solenoid. (Note, the magnetization of ferromagnetic materials SATURATES in a large enough field: for Iron it’s at around 2 Tesla.) A| Stainless steel (paramagnetic) ~ +1% increase B| Iron (ferromagnetic) ~ +10-100 times increase C| Copper (diamagnetic) ~ -10-4 decrease

Magnetic Susceptibility Magnetic susceptibility Χm paramagnet diamagnet Relative permeability Km permeability saturation

Reading Quiz 1 Three currents I1, I2, and I3 are directed perpendicular to the plane of this page as shown. The value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path is -μo2I1 What is the magnitude of the currents in I2 and I3? a. I2=0, I3 can be any value b. I2=0, I3 can only be zero c. I2=I1, I3 can be any value d. I2=2I1, I3 can be any value e. I2=3I1, I3 can be any value I1 I3 I2

Magnetism Exhibited by Materials Diamagnetism: (small) magnetic moment opposite to the external magnetic field Bapp is induced Any material – but shows only if non-paramagnetic repelled from region of large B Paramagnetism: magnetic moment of individual atoms become aligned parallel to the applied magnetic field Bapp transition element, rare earth, ... attracted toward region of large B Ferromagnetism: magnetic moment of individual atoms are already (partially) aligned in some direction even if Bapp=0 Fe, Ni,... Antiferromagnetism: like ferromagnetism except that alternating moments are (partially) aligned opposite to each other (when B=0) Mn, Cr,...

Paramagnetism and diamagnetism E-77

Hysteresis for a Ferromagnet Lack of retraceability shown is called hysteresis. Memory in magnetic disk and tape Alignment of magnetic domains retained in rock (cf. lodestones) Area enclosed in hysteresis loop Energy loss per unit volume hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, stronger “memory”) soft magnet: narrow hysteresis loop (easy to demagnetize,…)

Bar magnet approaches coil Induction v S N Bar magnet approaches coil Current induced in coil v N S Reverse poles of magnet (or direction of motion) Current in opposite direction N S Bar magnet stationary No induced current v S N Coil moving around bar magnet Same currents induced in coil (RELATIVE motion) What’s in common?: Change of Magnetic flux = EMF!

Induction: Coil and B Fields 6D04

Magnetic Flux 1 Wb = 1 T m2 Gauss’s Law for Magnetism (N turns) over closed surface (N turns)

Faraday’s Law of Induction The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time. where N Minus sign indicates the sense of EMF: Lenz’s Law RHR determines the direction for EMF Fingers curl along red EMF direction, thumb points (left – WHY??) along the INDUCED B field which happens ONLY if the EMF can drive a current around the loop. If the loop is just mathematical, or an insulator, then the EMF is still present, but no induced B. N 12

Faraday’s Law of Induction Minus sign indicates the sense of EMF: Lenz’s Law Faraday’s Law of Induction The induced EMF will tend to drive current in the loop (if the loop is conducting) to create extra B which OPPOSES the CHANGE in B. Opposes not B, but rather, the CHANGE in B In the example below, the original B is increasing to the right, so the induced current in a wire loop would generate extra B -- TO THE LEFT. B to the right but DECREASING would induce opposite EMF trying to create more B to the right to MAINTAIN the original B. N A magnetic field has a kind of “inertia” – a collapsing field tries to induce currents to prevent (retard) the collapse. And it also takes time and energy to build up a magnetic field. N

Warm-up quiz 2 The magnetic field shown into the figure is decreasing: what’s the direction of the induced current in the closed rectangular loop? Clockwise Counterclockwise No induced currents. Need induced EMF/ i tending to “make up for” the decreasing B into figure. Clockwise

Induced Electric Field from Faraday’s Law EMF is work done per unit charge: If work is done on charge q, electric field E must be present: Rewrite Faraday’s Law in terms of induced electric field: This form relates E and B! magnitude increasing Note that for E fields generated by charges at rest (electrostatics) since this would correspond to the potential difference between a point and itself. => Static E is conservative. The induced E by magnetic flux changes is non-conservative (nc).

Conducting Loop in a Changing Magnetic Field Induced EMF has a direction such that it opposes the change in magnetic flux that produced it. approaching moving away Magnetic moment μ created by induced currrent I attracts the bar magnet. Magnetic moment μ created by induced currrent I repels the bar magnet. Force on ring is attractive. Force on ring is repulsive.

Faraday Induction and i2R Losses 6D09

Faraday’s and Lenz’s Laws At 2, ϕB is increasing into page. So emf is induced to produce a counterclockwise current. At 4, ϕB in decreasing into page. So current is clockwise. At 1, 3, and 5, ϕB is not changing. So there is no induced emf.

Ways to Change Magnetic Flux Changing the magnitude of the field within a conducting loop (or coil). Changing the area of the loop (or coil) that lies within the magnetic field. Changing the relative orientation of the field and the loop. generator motor

Other Examples of Induction EMF induced in Coil 2 + - EMF is induced again + - Switch has been open for some time: Nothing happening Switch is just opened: Switch is just closed: EMF is induced in coil Self-induction in this case + - Switch is just closed: Back emf (counter emf)

9:30 Quiz 3 March 8, 2011 Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl clockwise around the circular path shown? a. 5μ0I b. −μ0I c. μ0I d. 6μ0I e. zero I 2I 3I

10:30 Quiz 3 March 8, 2011 A current directed toward the top of the page and a rectangular loop of wire lies in the plane of the page. Both are held in place by an external force. If the current I is decreasing, what is the direction of the magnetic force on the left edge of the loop? I a. Toward the right b. Toward the left c. Toward top of page d. Toward bottom of page e. No force acts on it.

11:30 Quiz 3 March 8, 2011 A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. If the current I is increasing, what happens to the loop? I a. The loop is pulled toward the top of the page b. The loop is pulled toward the current c. A clockwise current is induced in the loop. d. A counterclockwise current is induced in the loop. e. Nothing happens to the loop

Quiz B Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path shown? 2I a. 4μ0I b. -2μ0I c. 2μ0I d. 6μ0I e. zero 3I I

Quiz C Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the magnitude of the Ampere’s Law line integral of B∙ dl clockwise around the circular path shown? a. 5μ0I b. 3μ0I c. μ0I d. 6μ0I e. zero 3I I 2I