Lecture 17 1 Amperes Law in Magnetostatics BiotSavarts

Lecture 17 -1 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 17 -2 Magnetization and “Bound” Current in Matter • Strong externally applied field B app aligns the magnetic moments in matter. (M) Magnetization

Lecture 17 -3 Hysteresis for a Ferromagnet Lack of retraceability shown is called hysteresis. Ø Memory in magnetic disk and tape Ø Alignment of magnetic domains retained in rock (cf. lodestones) Area enclosed in hysteresis loop Energy loss per unit volume v hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, high memory) v soft magnet: narrow hysteresis loop (easy to demagnetize, …)

Lecture 17 -4 BRIDGE OF NAILS

Lecture 17 -5 INDUCTION • Bar magnet approaches coil S N v Current induced in coil • Reverse poles of magnet N S Current in opposite direction N • Bar magnet stationary v S No induced current v • Coil moving around bar magnet Same currents induced in coil S N What’s in common? : Change of Magnetic flux = EMF!

Lecture 17 -6 Magnetic Flux (N turns)

Lecture 17 -7 MAGNETIC BRAKING 6 D 08

Lecture 17 -8 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. N Minus sign indicates the sense of EMF: Lenz’s Law • Decide on which way n goes Fixes sign of ϕB • RHR determines the positive direction for EMF N

Lecture 17 -9 Induced Electric Field from Faraday’s Law Rewrite Faraday’s Law in terms of induced electric field: This form relates E and B! • 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.

Lecture 17 -10 JUMPING JACK 6 D 11

Lecture 17 -11 How to use Faraday’s law to determine the induced current direction 1. define the direction of normal direction, e. g. 2. determine the sign of Φ. Here Φ>0 ; can be any of the two point to right N 3. determine the sign of ∆Φ. Here ∆Φ >0 4. determine the sign of Δϕ ind using faraday’s law. Here Δϕ ind <0 5. RHR determines the positive direction for EMF 1. If Δϕ >0, current follow the direction of the curled fingers. 2. If Δϕ <0, current goes to the opposite direction of the curled fingers.

Lecture 17 -12 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 Ø Magnetic moment created by induced currrent I repels the bar magnet. Force on ring is repulsive. moving away Ø Magnetic moment created by induced currrent I attracts the bar magnet. Force on ring is attractive.

Lecture 17 -13 Faraday’s and Lenz’s Laws Ø At 1, 3, and 5, ΦB is not changing. So there is no induced emf. Ø At 4, ΦB in decreasing into page. So current is Ø At 2, ΦB is increasing into page. So clockwise. emf is induced to produce a counterclockwise current.

Lecture 17 -14 Motional EMF of Sliding Conductor Induced EMF: Ø Lenz’s Law gives direction ØFaraday’s Law Ø FM decelerates the bar Ø This EMF induces current I Ø Magnetic force FM acts on this I

Lecture 17 -15 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. motor generator

Lecture 17 -16 Other Examples of Induction + - Switch has been open for some time: Switch is just closed: Nothing happening EMF induced in Coil 2 + - Switch is just opened: EMF is induced again Switch is just closed: EMF is induced in coil - + Back emf (counter emf)