Chapter 29: Mastering Physics

PhET Tutorial: Faraday's Electromagnetic Lab Faraday's law of induction deals with how a magnetic flux induces an emf in a circuit. Recall that magnetic flux depends on magnetic field strength and the effective area the field is passing through. We'll start our investigation by looking at the field strength around a bar magnet. Select the Bar Magnet tab. Deselect Show Compass, and drag the bar magnet to the center of the screen. Then, select Show Field Meter, which allows you to measure the strength of the magnetic field at any location.

Select the Pickup Coil tab, and place the bar magnet inside the coil containing two loops. Try to find a location where the stationary magnet induces a current in the coil and causes the light bulb to shine. Which of the following is correct?

Now, let's look at a situation with changing flux. Starting from the far left of the screen, move the magnet to the right so it goes through the middle of the current loop at a constant speed and out to the right of the loop. Roughly where is the magnet when the light bulb is the brightest? (The brightness of the light bulb is depicted by the length of the rays emanating from it.)

How does the brightness of the light bulb change if the magnet is moved through the coil more quickly?

Select the Transformer tab, which contains an electromagnet at left (containing a DC battery) and a pickup coil attached to a light bulb at right. The electromagnet produces a magnetic field very similar to that of the bar magnet. Try to find a location where the stationary electromagnet (with a DC current) induces a current in the pickup coil and causes the light bulb to shine. Which of the following is correct?

Now, select an AC current source at top right (instead of the DC battery). You can adjust the frequency of oscillation of the current using the horizontal slider bar (in the blue AC Current Supply window), and you can adjust the peak current with the vertical slider bar. Place the electromagnet to the left of the pickup coil (as shown below). How does the average brightness of the light bulb depend on the AC frequency?

With the electromagnet still situated just to the left of the pickup coil, turn the frequency all the way down to its lowest setting (5%). Watch how the brightness changes in time and how it correlates with the instantaneous current (which is shown by the position of the red vertical line in the AC current supply meter graph). When is the brightness the greatest?

Adjust the frequency to 100% so the bulb is always "on." With the electromagnet still just to the left of the pickup coil, how does the average brightness of the light bulb depend on the number of loops of the pickup coil? (You can change the number of loops in the Pickup Coil properties box at bottom right.)

With the electromagnet still just to the left of the pickup coil, how does the average brightness of the light bulb depend on the loop area of the pickup coil? (You can change the loop area (as a percentage) using the slider bar at bottom right.)

Now, place the electromagnet centered inside the pickup coil, as shown below. How does the average brightness of the light bulb depend on the loop area of the pickup coil?

± Motional EMF in a Conducting Rod In the figure, a conducting rod with length L = 25.0 cm moves in a magnetic field B⃗ of magnitude 0.500 T directed into the plane of the figure. The rod moves with speed v = 7.00 m/s in the direction shown. (Figure 1)

When the charges in the rod are in equilibrium, which point, a or b, has an excess of positive charge?

In what direction does the electric field then point?

When the charges in the rod are in equilibrium, what is the magnitude E of the electric field within the rod? Express your answer in volts per meter to at least three significant digits.

Which point, a or b, is at higher potential?

What is the magnitude Vba of the potential difference between the ends of the rod? Express your answer in volts to at least three significant digits.

What is the magnitude E of the motional emf induced in the rod? Express your answer in volts to at least three significant digits.

Induced Current in a Metal Loop Conceptual Question For each of the actions depicted below, a magnet and/or metal loop moves with velocity v⃗ (v⃗ is constant and has the same magnitude in all parts). Determine whether a current is induced in the metal loop. If so, indicate the direction of the current in the loop, either clockwise or counterclockwise when seen from the right of the loop. The axis of the magnet is lined up with the center of the loop.

For the action depicted in the figure, (Figure 1) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

For the action depicted in the figure, (Figure 2) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

For the action depicted in the figure, (Figure 3) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

For the action depicted in the figure, (Figure 4) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

For the action depicted in the figure, (Figure 5) indicate the direction of the induced current in the loop (clockwise, counterclockwise or zero, when seen from the right of the loop).

Faraday's Law and Induced Emf Learning Goal: To understand the terms in Faraday's law and to be able to identify the magnitude and direction of induced emf. Faraday's law states that induced emf is directly proportional to the time rate of change of magnetic flux. Mathematically, it can be written as E=−ΔΦBΔt, where E is the emf induced in a closed loop, and ΔΦBΔt is the rate of change of the magnetic flux through a surface bounded by the loop. For uniform magnetic fields the magnetic flux is given by ΦB=B⃗ ⋅A⃗ =BAcos(θ), where θ is the angle between the magnetic field B⃗ and the normal to the surface of area A. To find the direction of the induced emf, one can use Lenz's law: The induced current's magnetic field opposes the change in the magnetic flux that induced the current. For example, if the magnetic flux through a loop increases, the induced magnetic field is directed opposite to the "parent" magnetic field, thus countering the increase in flux. If the flux decreases, the induced current's magnetic field has the same direction as the parent magnetic field, thus countering the decrease in flux. Recall that to relate the direction of the electric current and its magnetic field, you can use the right-hand rule: When the fingers on your right hand are curled in the direction of the current in a loop, your thumb gives the direction of the magnetic field generated by this current.

Which of the following changes would induce an electromotive force (emf) in the loop? When you consider each option, assume that no other changes occur. Check all that apply.

Find the flux ΦB through the loop. Express your answer in terms of x, y, and B.

If the magnetic field steadily decreases from B to zero during a time interval t, what is the magnitude E of the induced emf? Express your answer in terms of x, y, B, and t.

If the magnetic field steadily decreases from B to zero during a time interval t, what is the magnitude I of the induced current? Express your answer in terms of x, y, B, t, and the resistance R of the wire.

If the magnetic field steadily decreases from B to zero during a time interval t, what is the direction of the induced current?

Which of the following changes would result in a clockwise emf in the loop? When you consider each option, assume that no other changes occur. Check all that apply.

Induced EMF and Current in a Shrinking Loop Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 163 cm , but its circumference is decreasing at a constant rate of 13.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.800 T , which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop.

Find the magnitude of the emf E induced in the loop after exactly time 3.00 s has passed since the circumference of the loop started to decrease. Express your answer numerically in volts to three significant digits.

Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

± Magnetic Flux and Induced EMF in a Coil In a physics laboratory experiment, a coil with 160 turns enclosing an area of 13.3 cm2 is rotated during the time interval 4.90×10−2 s from a position in which its plane is perpendicular to Earth's magnetic field to one in which its plane is parallel to the field. The magnitude of Earth's magnetic field at the lab location is 5.30×10−5 T .

What is the total magnitude of the magnetic flux ( Φinitial) through the coil before it is rotated? Express your answer numerically, in webers, to at least three significant digits.

What is the magnitude of the total magnetic flux Φfinal through the coil after it is rotated? Express your answer numerically, in webers, to at least three significant digits.

What is the magnitude of the average emf induced in the coil? Express your answer numerically (in volts) to at least three significant digits.

Faraday's Law and Electric Fields Learning Goal: To understand the terms in Faraday's law for magnetic induction of electric fields, and contrast these fields with those produced by static charges. Faraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. It relates the line integral of the electric field around a closed loop to the change in the total magnetic field integral across a surface bounded by that loop: E=−dΦMdt=−ddt∫B⃗ (r⃗ ,t)⋅dA⃗ , where E=∮E⃗ (r⃗ ,t)⋅dl⃗ is the line integral of the electric field, and the magnetic flux is given by ΦM=∫B⃗ (r⃗ ,t)⋅dA⃗ =∫B|dA|cos(θ), where θ is the angle between the magnetic field B⃗ and the local normal to the surface bounded by the closed loop. (Figure 1) Direction: The line integral and surface integral reverse their signs if the reference direction of dl⃗ or dA⃗ is reversed. The right-hand rule applies here: If the thumb of your right hand points along dA⃗ , then the fingers point along dl⃗ . You are free to take the loop anywhere you choose, although usually it makes sense to choose it to lie along the path of the circuit you are considering.

Consider the direction of the electric field in the figure. Assume that the magnetic field points upward, as shown. Under what circumstances is the direction of the electric field shown in the figure correct?

Now consider the magnetic flux through a surface bounded by the loop. Which of the following statements about this surface must be true if you want to use Faraday's law to relate the magnetic flux to the line integral of the electric field around the loop?

When can an electric field be measured at any point from the force on a stationary test charge at that point?

When can an electric field that does not vary in time arise?

When will the integral ∮E⃗ ⋅dl⃗ around any closed loop be zero?

A cylindrical iron rod of infinite length with cross-sectional area A is oriented with its axis of symmetry coincident with the z axis of a cylindrical coordinate system as shown in the figure. It has a magnetic field inside that varies according to Bz(t)=B0+B1t. Find the theta component Eθ(R,t) of the electric field at distance R from the z axis, where R is larger than the radius of the rod. (Figure 2) Express your answer in terms of A, B0, B1, R, and any needed constants such as ϵ0, π, and μ0.

Inductance of a Coaxial Cable A coaxial cable consists of alternating coaxial cylinders of conducting and insulating material. Coaxial cabling is the primary type of cabling used by the cable television industry and is also widely used for computer networks such as Ethernet, on account of its superior ability to transmit large volumes of electrical signal with minimum distortion. Like all other kinds of cables, however, coaxial cables also have some inductance that has undesirable effects, such as producing some distortion and heating.

Consider a long coaxial cable made of two coaxial cylindrical conductors that carry equal currents I in opposite directions (see figure). The inner cylinder is a small solid conductor of radius a. The outer cylinder is a thin walled conductor of outer radius b, electrically insulated from the inner conductor. Calculate the inductance per unit length Ll of this coaxial cable. (Figure 1) ( L is the inductance of part of the cable and l is the length of that part.) Due to what is known as the "skin effect", the current I flows down the (outer) surface of the inner conducting cylinder and back along the outer surface of the outer conducting cylinder. However, you may ignore the thickness of the outer cylinder. Express your answer in terms of some or all the variables I, a, b, and μ0, the permeability of free space.

± Basic Properties of Inductors Learning Goal: To understand the units of inductance, the potential energy stored in an inductor, and some of the consequences of having inductance in a circuit. After batteries, resistors, and capacitors, the most common elements in circuits are inductors. Inductors usually look like tightly wound coils of fine wire. Unlike capacitors, which produce a physical break in the circuit between the capacitor plates, the wire of an inductor provides an unbroken continuous path in which current can flow. When the current in a circuit is constant, an inductor acts essentially like a short circuit (i.e., a zero-resistance path). In reality, there is always at least a small amount of resistance in the windings of an inductor, a fact that is usually neglected in introductory discussions. Recall that current flowing through a wire generates a magnetic field in the vicinity of the wire. If the wire is coiled , such as in a solenoid or an inductor, the magnetic field is strongest within the coil parallel to its axis. The magnetic field associated with current flowing through an inductor takes time to create, and time to eliminate when the current is turned off. When the current changes, an EMF is generated in the inductor, according to Faraday's law, that opposes the change in current flow. Thus inductors provide electrical inertia to a circuit by reducing the rapidity of change in the current flow.

Based on the equation given in the introduction, what are the units of inductance L in terms of the units of E, t, and I (respectively volts V, seconds s, and amperes A)?

What EMF is produced if a waffle iron that draws 2.5 amperes and has an inductance of 560 millihenries is suddenly unplugged, so the current drops to essentially zero in 0.015 seconds? Express your answer in volts to two significant figures.

Which of the following changes would increase the potential energy stored in an inductor by a factor of 5? Check all that apply.

Which of the graphs illustrate how the current through an inductor might possibly change over time?(Figure 1) Type the numbers corresponding to the right answers in alphabetical order. Do not use commas. For instance, if you think that only graphs C and D are correct, type CD.

± Energy Stored in an Inductor The electric-power industry is interested in finding a way to store electric energy during times of low demand for use during peak-demand times. One way of achieving this goal is to use large inductors. What inductance L would be needed to store energy E=3.0kWh (kilowatt-hours) in a coil carrying current I=300A?