Exploratorial.004 - Faraday's Law


(a) (i) A single circular coil of radius 3cm is oriented in a uniform magnetic field B = -0.10 T such that the plane of the coil is perpendicular to the magnetic field (see Fig 1.). What is the value of the magnetic flux F through the coil?
(ii) The coil of part (i) is now re-oriented so that the plane of the coil is at angle Q = 45 to the direction of the magnetic field. What is the value of the magnetic flux through the coil in this position?
(iii) What orientation of the coil will give F = 0 ?

Fig 1. (a): Left, perspective view: Arrows indicate the direction of the magnetic field. We see the field is uniform because the lines are straight and evenly spread out.
Right, plan view: By convention, a magnetic field into the page is represented by crosses (like the view of the end of an arrow travelling away from us).
(b): coil plane at 45° to the magnetic field.

(b) A strong permanent magnet is moved towards and away from a 4 cm diameter circular coil consisting of ten turns of wire, as shown in Fig 2. The magnet is moved at constant speed. In its initial and final positions the magnet is far enough away from the coil that the magnetic field produced at the coil position is essentially zero.

  1. When the magnet is moved towards the coil the magnetic field over the coil area increases from B1 = 0 to B2 = 0.80 T in a time Dtin = 0.25 s. What is the magnitude of the e.m.f. induced in the coil and in which direction would a current flow in the coil if the ends of the coil were connected to form a circuit?
  2. When the magnet is moved away from the coil the magnetic field reduces from B2 = 0.80 T to B3 = 0 T in a time Dtout = 0.1 s. What is magnitude of the e.m.f. induced in the coil and in which direction (clockwise or anticlockwise when viewed from the magnet side) would a current flow in the coil if the ends of the coil were connected to form a circuit?

Fig 2. A strong permanent magnet is moved toward and away from a circular coil of diameter 4 cm. The coil consists of 10 turns of wire.

  • Initially, the magnet is sufficiently far from the coil that we can approximate the magnetic field over the area of the coil to be B = B1 = 0.
  • The magnet is moved into the coil, B = B2 = 0.80 T (maximum B), and momentarily stationary before moving out.
  • Magnet is moved away from the coil at constant speed finishing far from the coil where B = B3 = 0 again.

(c) The circular coil described in (a) above is now mounted on an axle which runs across a diameter of the coil (see Fig. 3, below). Magnets positioned as shown below provide a uniform magnetic field B = 0.25 T across the region of the coil. The coil is rotated at a constant rate of 50 revolutions per second. This is a generator. You will find useful information in the course text, Physics, 5th Ed., Giancoli, Section 21-5, p629. There are also very nice, instructive java applets at
http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/generatorengl.htm and

(i) Find the expression for the emf generated across the terminals of the rotating coil as a function of time which contains the coil parameters (area, number of windings), the magnetic field B and the angular frequency w.
(ii) What are the values of the frequency f and the angular frequency w of variation of the emf ?
(iii) Sketch a graph of the output emf of the coil as a function of time. Label the graph axes using the values you calculated above.

Hands-on activities in the lab

Part1: Induced voltages in circular loops of wire.

You are provided with three coils having different diameters (see Fig. 4 above) and a strong permanent magnet. The output of the coils can be displayed on a digital voltmeter or on an oscilloscope, see Fig. 5, below.

Use these coils to investigate how induced voltages are produced by a changing magnetic flux. In particular, test how the induced voltage varies with:

(i) coil diameter:

(ii) the rate of change of flux:

Also investigate how the polarity of the induced voltage (is it +ve or –ve) varies with:
(iii) the direction of the magnetic field (i.e. a N pole or a S pole is brought up to the coil):

(iv) direction of travel of magnet (i.e. magnet towards or away from the coil):

(v) which side of the coil faces the magnet:


Part 2: Building a motor and electrical generator

We have provided you with a commercially made miniature electric motor in a plastic support frame with gear wheels as shown below. This motor is operated from a battery.

You also have an electric generator in kit form (shown below) which you will assemble and connect to the miniature motor through the gear assembly; the miniature motor therefore ‘powers’ (turns) the generator. (Note: this generator kit is actually a motor kit! Here we are running the motor as a generator. You will see in this Workshop that motors and generators are closely related.)


Comparing this (actual) generator to the schematic generator discussed earlier in the presentation we can identify the main features.

The rectangular coil of N turns rotates in the field B at w radians per second. The coil area is A (= length × width) square metres.

The emf generated is given by
e = NAB w sinwt (Volts)

The generator output, Vout, is an alternating (AC) voltage of amplitude V0 = NABw and frequency w (= 2pf) radians per second.

Connect your assembled generator to the motor as shown below.

Verify that the generator is producing an output by displaying the output voltage on the multimeter.


(a) A circular coil consisting of a single turn of wire is positioned in a uniform magnetic field of strength 0.5 T. Find the magnetic flux F through the coil when the coil is oriented with its plane at the following angles to the field: (i) 0o, (ii) 30o, (iii) 45o, (iv) 90o.
(b) A generator consists of a square coil of side 4 cm with a total of 75 windings. The coil rotates in a uniform field of strength 0.25 T and a rate of 50 rotations per second. What is the output voltage of this generator?
(c) A car’s alternator (so named because it produces an alternating output voltage) is very similar in operating principle to the AC generator you have constructed in the class. If an alternator produces an output voltage of 15v when the engine is ticking over at 800 rpm (revs per minute), what will be the output of the alternator at an engine speed of 3500 rpm. (For this estimate, assume that the alternator speed is the only thing that changes).
(d) You decide to measure the strength of the Earth’s magnetic field by rotating a coil connected to a voltmeter in the Earth’s magnetic field. Your coil is 20 cm in diameter and consists of 500 turns of thin copper wire. Assuming the Earth’s field is pointing directly downwards you align the coil appropriately and rapidly rotate (twist) the coil by 90o in a time 0.1 secs. If the Earth’s field is Bearth = 0.5xF-4T what will be the measured voltage?

The Earth’s magnetic field has components in the horizontal and vertical directions so the question is a simplification of the real situation.

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Last Updated 13th August 2005