Faraday's Law

  • The induced emf due to a changing magnetic field is equal in magnitude to the rate of change of the magnetic flux through a surface bounded by the circuit

  • The direction of the induced current is given by Lenz's Law

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    Faraday's

Lenz's Law

  • The current induced by a changing magnetic flux creates a magnetic field opposing the change in flux

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Maxwell's Equations

• • • • Gauss's Law Gauss's Law for Magnetism enc Faraday's Law mpere's Law É.di= closed loop fj.dA dt open surface closed loop

Example 1: Induced Current in a Loop

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  1. Find the generated emf as a function of time

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  2. Determine the current through the 100-ohm lamp as a function of time

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  3. What is the direction of the current through the loop at time t=5s?

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    • Clockwise, due to Lenz's Law

Example 2: Rod on Rails

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  • Consider a circuit in which a current-carrying rod on rails is moved to the left with constant velocity v. If the circuit is perpendicular to a constant magnetic field, determine the induced emf in the circuit

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2012 Free Response Question 3

Crossbar X x x x X X X X x x x x X X 130 X X x X x x x X x X x X x X x x x X X X X x X X x X X X X A closed loop is made of a U-shaped metal wire of negligible resistance and a movable metal crossbar of resistance R. The crossbar has mass m and length L It is initially located a distance ho from the other end of the loop. The loop is placed vertically in a uniform horizontal magnetic field of magnitude Bo in the direction shown in the figure above. Express all algebraic answers to the questions below in terms of Bo , L, m, ho , R, and fundamental constants, as appropriate. (a) Determine the magnitude of the magnetic flux through the loop when the crossbar is in the position shown. The crossbar is released from rest and slides with negligible friction down the U-shaped wire without losing electrical contact. (b) On the figure below, indicate the direction of the current in the crossbar as it falls. Justify your answer. (c) Calculate the magnitude of the current in the crossbar as it falls as a function of the crossbar's speed v .

6) c) -4(βοΙΟ\_ Ι30Ιν

(d) Derive, but do NOT solve, the differential equation that could be used to determine the speed v of the crossbar as a function of time t. (e) Determine the terminal speed VT of the crossbar. (D If the resistance R of the crossbar is increased, does the terminal speed increase, decrease, or remain the same? Increases Decreases Remains the same Give a physical justification for your answer in terms of the forces on the crossbar.

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2010 Free Response Question 3

Lightbulb (resistance R) E\&M. 3. The long straight wire illustrated above carries a current I to the right. The current varies with time t according to the equation I = 10 — Kt , where 10 and K are positive constants and I remains positive throughout the time period of interest. The bottom of a rectangular loop of wire of width b and height a is located a distance d above the long wire, with the long wire in the plane of the loop as shown. A lightbulb with resistance R is connected in the loop. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Indicate the direction of the current in the loop. Clockwise Justify your answer. Counterclockwise (b) Indicate whether the lightbulb gets brighter, gets dimmer, or stays the same brightness over the time period of interest. Gets brighter Justify your answer. Gets dimmer Remains the same (c) (d) (e) Determine the magnetic field at t = O due to the current in the long wire at distance r from the long wire. Derive an expression for the magnetic flux through the loop as a function of time. Derive an expression for the power dissipated by the lightbulb.

SqomN (2 ,m22 (b

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2009 Free Response Question 3

x x x x E\&M. 3. x x 2 x x A square conducting loop of side L contains two identical lightbulbs, 1 and 2, as shown above. There is a magnetic field directed into the page in the region inside the loop with magnitude as a function of time t given by B (t) = at + b , where a and b are positive constants. The lightbulbs each have constant resistance Ro . Express all answers in terms of the given quantities and fundamental constants. (a) (b) (c) Derive an expression for the magnitude of the emf generated in the loop. i. Determine an expression for the current through bulb 2. ii. Indicate on the diagram above the direction of the current through bulb 2. Derive an expression for the power dissipated in bulb 1.

& \_ 23 (p

Another identical bulb 3 is now connected in parallel with bulb 2, but it is entirely outside the magnetic field, as shown below. x x 1 x x B x x 2 x x 3 (d) How does the brightness of bulb 1 compare to what it was in the previous circuit? Brighter Justify your answer. Dimmer The same

Now the portion of the circuit containing bulb 3 is removed, and a wire is added to connect the midpoints of the top and bottom of the original loop, as shown below. (e) How does the brightness of bulb 1 compare to what it was in the first circuit? Brighter Justify your answer. Dimmer The same

, , 8 冖 P

2008 Free Response Question 3

1 Figure 1 E\&M. 3. The circular loop of wire in Figure 1 above has a radius of R and carries a current I. Point P is a distance of R/2 above the center of the loop. Express algebraic answers to parts (a) and (b) in terms of R, I, and fundamental constants. (a) i. State the direction of the magnetic field Bl at point P due to the current in the loop. ii. Calculate the magnitude of the magnetic field Bl at point P.

503 R

1 Figure 2 Axis Figure 3 A second identical loop also carrymg a current I is added at a distance of R above the first loop, as shown in Figure 2 above. (b) Determine the magnitude of the net magnetic field Bnet at point P. A small square loop of wire in which each side has a length s is now placed at point P with its plane parallel to the plane of each loop, as shown in Figure 3 above. For parts (c) and (d), assume that the magnetic field between the two circular loops is uniform in the region of the square loop and has magnitude B net • (c) In terms of Bnet and s, determine the magnetic flux through the square loop. (d) The square loop is now rotated about an axis in its plane at an angular speed O. In terms of Bnet , s, and o , calculate the induced emf in the loop as a function of time t, assuming that the loop is horizontal at t = O.

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2007 Free Response Question 3

x x x x x x x x x x x x x x x x x x x x x x x x x x Top View x x x x x x x x x x x x x x x x x x In the diagram above, a nichrome wire of resistance per unit length is bent at points P and Q to form horizontal conducting rails that are a distance L apart. The wire is placed within a uniform magnetic field of magnitude B pointing into the page. A conducting rod of negligible resistance, which was aligned with end PQ at time t = O, slides to the right with constant speed v and negligible friction. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Indicate the direction of the current induced in the circuit. Clockwise Justify your answer. Counterclockwise (b) Derive an expression for the magnitude of the induced current as a function of time t. (c) Derive an expression for the magnitude of the magnetic force on the rod as a function of time.

2(l+2ve) c) βιν (Ι+Ζνέ) 3(Ι+Ζνέ)

(d) On the axes below, sketch a graph of the external force Fext as a function of time that must be applied to the rod to keep it moving at constant speed while in the field. Label the values of any intercepts. ext Time (e) The force pulling the rod is now removed. Indicate whether the speed of the rod increases, decreases, or remains the same. Increases Justify your answer. Decreases Remains the same

s•soddo ON ( 2

2006 Free Response Question 3

Spring x x x x x x x x x x x x s x x x x Magnetic Field A loop of wire of width w and height h contains a switch and a battery and is connected to a spring of force constant k, as shown above. The loop carries a current I in a clockwise direction, and its bottom is in a constant, uniform magnetic field directed into the plane of the page. (a) On the diagram of the loop below, indicate the directions of the magnetic forces, if any, that act on each side the loop. (b) The switch S is opened, and the loop eventually comes to rest at a new equilibrium position that is a distance x from its former position. Derive an expression for the magnitude Bo of the uniform magnetic field in terms of the given quantities and fundamental constants.

Кх

The spring and loop are replaced with a loop of the same dimensions and resistance R but without the battery and switch. The new loop is pulled upward, out of the magnetic field, at constant speed vo. Express algebraic answers to the following questions in terms of B D R, and the dimensions of the loop. (c) i. On the diagram of the new loop below, indicate the direction of the induced current in the loop as the loop moves upward. x x x x x x X x x x x x x x x x ii. Derive an expression for the magnitude of this current. (d) Derive an expression for the power dissipated in the loop as the loop is pulled at constant speed out of the field. (e) Suppose the magnitude of the magnetic field is increased. Does the external force required to pull the loop at speed vo increase, decrease, or remain the same? Increases Justify your answer. Remains the same

Ź = -Bvvo -49, Bwvo

2013 Free Response Question 3

x x x x x X x X x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x The figure above shows a circular loop of area 0.25 m2 and resistance 12 Q that lies in the plane of the page. A magnetic field of magnitude B directed into the page exists in the area of the loop. The field varies with time t, as shown in the graph below. 1.8 1.2 - 0.6 o B = 1.8e-0.05r 4 8 12 16 18 (a) . Derive an expression for the magnitude of the induced emfin the loop as a function of time for the interval t = O s to t = 8 s. ii. Calculate the magnitude of the induced current I in the loop at time t = 4 s.

hS 1000 as 220' 3 s zoo — = • OVI)

(b) i. Sketch a graph of the induced current Iin the loop as a function of time t from t = O s to t = 18 s on the axes below, assummg that a counterclockwise (CCW) current is positive. ccw O cw 4 8 12 16 18 ii. For the time interval 12 s to 16 s, justify the direction of the current you have indicated in your graph. (c) Calculate the total energy dissipated in the loop during the first 8 s shown.

4022.) T T e S -%ZZ.IO e

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