Faraday s Law of Electromagnetic Induction 2.1 Represent and reason The rectangular loop with a resistor is pulled at constant velocity through a uniform external magnetic field that points into the paper in the regions shown with the crosses (x). Complete the table that follows to determine qualitatively the shape of the induced current-versus-time graph. Draw a qualitative fluxversus-time graph for the process (positive direction in and negative direction out). Draw a qualitative induced magnetic fieldversus-time graph for the process. Draw a qualitative induced currentversus-time graph for the process. Flux Φ Induced magnetic field B ind Induced current I t 1 t 2 t 3 t 4 t 1 t 2 t 3 t 4 t 1 t 2 t 3 t 4 Time t Time t Time t
2.2 Observe and explain In the table below, the results of four experiments are shown in which a changing magnetic field produced by an electromagnet passes through a loop. This changing B field causes a changing flux Φ through the loop and an induced current i ind around the loop of resistance R. The product i ind R is also plotted as a function of time. (a) Draw another graph in the table cells that shows the induced emf ε ind in the loop. Coil resistance is 1.0 Ω. Coil resistance is 3.0 Ω. i i ind (A) i ind (A) 0.10 i ind R (A Ω) i ind R (A Ω) Coil resistance is 2.0 Ω. Coil resistance is 6.0 Ω. i in (A) i in (A) i in R (A Ω) 0.10 i in R (A Ω) 0 0 (b) Devise a relationship between ΔΦ/Δt and ε ind. Do not forget the sign!
2.3 Observe and explain The analysis of the following experiment provides an expression for the potential difference produced in a loop moving into, through, and out of a magnetic field. This is called motional emf. Experiment Analysis Analysis A metal bar of length L moves at constant velocity through a magnetic field that points into the paper (the crosses). (a) Indicate on the bar how the magnetic force of the field on charges in the bar redistributes charge in the bar. (b) This charge redistribution, which occurs quickly, produces an electric field inside the bar that prevents further charge redistribution. Draw the E-field lines. (c) Apply Newton s second law for a charge in the (d) Use the expression that relates electric field E y middle of the bar now in an E field and a B field. and the potential difference over a distance V/L with the previous results to determine an expression for the potential difference (emf) across the ends of the bar. (e) Below, you see a rectangular metal conductor entering the magnetic field, completely in the magnetic field, and leaving the magnetic field described above. Use the results of (a) and (b) to draw charge distributions on the front and back vertical parts of the rectangle due to the force of the magnetic field on charge in the metal. Note: The magnetic field only exerts forces on charge in the metal parts that are in the field, not on those outside the fields. Will current flow? If so, indicate the direction. (f) Use the results of (d) to write an expression for the potential difference induced around the rectangular metal conductor while entering the field. (g) Use the results of (d) to write an expression for the potential difference induced around the rectangular metal conductor when completely in the field. Note that there are now charge distributions that cancel. (h) Use the results of (d) to write an expression for the potential difference induced around the rectangular metal conductor while leaving the field.
2.4 Observe and explain We repeat the previous activity, only this time using ideas of flux and induced emf. When you are finished, check to see if the results are consistent with those in Activity 2.3. The same rectangular metal conductor as in Activity 2.3e is entering the magnetic field, completely in the magnetic field, and leaving a magnetic field that points into the paper. (a) Draw a graph showing the magnetic flux through the opening of the metal conductor as a function of time while the rectangle is entering the magnetic field, completely in the magnetic field, and leaving the magnetic field. Then use the flux graph to make a graph of the induced emf for the same time interval. Φ t ε in Entering In field Leaving t (b) Write an expression for the changing flux as the rectangle enters the field. Then use this expression to determine the emf while the rectangle is entering the field. (c) Write an expression for the changing flux as the rectangle is completely in the field. Then use this expression to determine the emf while the rectangle is in the field. (d) Compare the expressions in (b) and (c) with the expressions determined in 2.3 f and g. (We skip the calculation for when the rectangle is leaving the field it s a little more messy.) Did you know? Before moving forward, you should know the following: magnetic flux: Φ = AB cos θ (units: [T*m 2 ] = [Wb] Weber) Faraday s law of electromagnetic induction: The average magnitude of the induced emf ε ind in a coil with N loops is the magnitude of the ratio of the magnetic flux change in the loop ΔΦ to the time interval t during which that flux change occurred multiplied by the number N of loops: ε ind = N ΔΦ / Δt Lenz s law: the direction of an induced current is such that its induced magnetic field opposes the change in the external magnetic flux that caused the induced current.
2.5 Reason The magnitude of the magnetic field in each situation described below is 0.50 T. For each situation, in the table that follows write an expression for the magnetic flux through the loop. Situation Loop and B r in the plane of the paper. Write an expression for the flux Loop perpendicular to the paper and r B in plane of the paper Loop in the plane of the paper Square loop of side a in the plane of the paper. r B into the paper. x x x x x x x x 37 o x
2.6 Represent and reason Four situations are shown in which the external flux through a loop is plotted as a function of time. In the table that follows, draw another graph that shows the induced emf in the loop as a function of time. 0.4 T m 2 T m 2 0 s 1 s 2 s 3 s 0 s 1 s 2 s 3 s 4 s +0.4 + 0.4 0 s 1 s 2 s 3 s time 0 s 1 s 2 s 3 s 4 s time 0.4 T m 2 T m 2 0.15 0 s 1 s 2 s 2.5 s +0.4 0 s 1 s 2 s 3 s time + 0 s 1 s 2 s 3 s 4 s 0.4 0 s 1 s 2 s 3 s 4 s time
2.7 A horizontal bar is pulled at constant velocity through an inward-pointing magnetic field. The bar slides on two horizontal, frictionless metal rails moving away from a resistor connected between their ends. Derive an expression for the induced current through the resistor of resistance R in terms of any or all quantities that you choose to include in a sketch of this system. Be sure to identify the quantities in the sketch. a. Draw top view sketches showing the rails and the bar location at an initial time and at a later time. Include symbols for quantities involved in the problem. b. Describe the assumptions you are you making. c. Draw a flux vs clock reading graph. d. Draw a consistent emf-versus-clock reading graph e. Represent flux and emf mathematically. f. Combine the mathematical representation with Ohm s law to get the desired expression for the current. 2.8 Consider the situation above, but with the bar moving toward the resistor instead. An external force does work as the movable bar moves, and this work is converted to electrical energy. Because the circuit is in a magnetic field, the flux through it changes with time, inducing a current. a. What is the direction of the induced current in the resistor? b. If the bar is 20 cm long and is pulled at a steady speed of 10 cm/s, what is the induced current if the resistor has a value of 5.0 Ω and the circuit is in a uniform magnetic field of 0.25 T? Did you know? Motional emf: An electric potential difference is produced between two points that depends on the magnitude of the electric field, E, in the rod and the distance L between the two ends ε motional emf = EL = vbl For problems involving conducting objects moving in a magnetic field, we can use either Faraday s law or the motional emf expression to determine the emf produced. Both methods yield the same result.