Last time: how charged particles move in a magnetic field Last time: how charged particles move in a magnetic field. Consider a uniform magnetic field into the board, with conducting rod moving through it: + side 𝑣 Charges will move If part of a circuit, can generate a current! - side

Which direction is the current through the resistor? Up Down Note: as rod moves, there is an increasing magnetic flux through the loop.

Magnetic Flux Just like for electric fields, can define magnetic flux through surface: Φ 𝐵 = 𝐵 ⋅𝑑 𝐴 For uniform 𝐵 and a flat surface 𝐴 : Φ 𝐵 =𝐵𝐴 cos 𝜃

Faraday’s Law Changing magnetic field can induce a ℇ and 𝐼: ℇ=− 𝑑 Φ 𝐵 𝑑𝑡 where Φ 𝐵 is the magnetic flux. - Solenoid with alternating / direct current. - Vary number of windings.

What are ways that we can increase the amount of current through the loop? More windings? Angle of loop relative to solenoid? Shape of loop?

Top View Solenoids B B Wire Loops Magnetic field ~inside of solenoid only  area is the same for both  flux is the same for both.

1. bulb A goes out; bulb B gets brighter. In figure (a), a solenoid produces a magnetic field whose strength increases into the plane of the page. An induced emf is established in a conducting loop surrounding the solenoid, and this emf lights bulbs A and B. In figure (b), points P and Q are shorted. After the short is inserted, 1. bulb A goes out; bulb B gets brighter. 2. bulb B goes out; bulb A gets brighter. 3. bulb A goes out; bulb B gets dimmer. 4. bulb B goes out; bulb A gets dimmer. 5. both bulbs go out. 6. none of the above Answer: 1. Although one might expect both bulbs to go out when shorted, only bulb A does. Together with the original loop, the extra wire between P and Q creates a total of three loops (see following figure): (i) the original circular loop; (ii) the heavy loop consisting of the extra wire and the right half of the original loop; and (iii) the crescent-shaped conducting loop consisting of the extra wire plus the dashed wire connected to bulb A. Because there is no changing flux through loop (iii), bulb A will not light. Loop (ii), however, encloses a changing magnetic flux and thus bulb B lights. Loop (i) also encloses a changing magnetic flux, but it is “easier” for the electrons to flow around loop (ii) because the extra wire has zero resistance, whereas the dashed wire containing bulb A has finite resistance. Since the net resistance in the circuit is now less than it was before the wire was added, bulb B gets brighter.

Lenz’s Law The induced current creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop. ℇ=− 𝑑 Φ 𝐵 𝑑𝑡 Way to interpret the minus sign in Faraday’s Law

The current through the wire is decreasing and the loop is fixed relative to the wire. In which direction is the induced current in the loop? Clockwise Counter-clockwise No current is induced

The loop shown below is traveling towards the wire with the velocity shown. The current, 𝐼, is constant. In which direction is the induced current in the loop? Clockwise Counter-clockwise No current is induced

The loop shown below is traveling towards the right at a constant distance from the bottom wire. The current, 𝐼, is constant. In which direction is the induced current in the loop? Clockwise Counter-clockwise No current is induced 𝑣

Cool Application: Generators and Motors

A uniform magnetic field can produce an EMF in a conducting loop if: The field changes in magnitude. The loop changes position within the field. The loop rotates within the field. The loop is removed from the field. The loop changes size. All of the above All except 2) All except 3) All except 4)

Relation to Electric Fields Electric fields cause charges to move  induced EMF can be related to 𝐸 : 𝐸 ⋅𝑑 𝑠 =− 𝑑 Φ 𝐵 𝑑𝑡 where integral is a closed path. Note: Not necessary for a loop to be there (i.e., test charges) for an 𝐸 field to be created.

Eddy Currents Plate moving between poles of magnet: Force tends to damp motion of the plate.

Adding slots  fewer paths for current  higher resistance  less force.

Question: Let's say you take an ordinary wire coathanger and straighten out the hook shaped part that normally hangs over the coat rack. Now, you can spin the (roughly) triangular part around by twisting the straightened part between your fingers. Estimate the EMF that you can generate by spinning the hanger in the Earth's magnetic field (about 5.3 x 10⁻⁵ T). Eps = BA (theta dot)

Lab: Magnetic Field of a Slinky: - Magnetic field sensor w/ LoggerPro - Do theoretical predictions about magnetic field hold?

Quantitative Question (31.12): A coil of 15 turns and radius 10.0 cm surrounds a long solenoid of radius 2.00 cm and 1000 turns/meter. The current in the solenoid changes as 𝐼=5 sin 120𝑡 𝐴 , where 𝐼 is in amperes and t is in seconds. Find the induced emf in the 15-turn coil as a function of time.