Finally! Flux! Electromagnetic Induction

Objectives

A wire moving in a magnetic field Imagine a wire of length L is moved with velocity v in a region of a magnetic field of constant magnitude B. Assume for convenience that B is coming out of the page and the wire moves from top to bottom. The wire is conducting (meaning it has many free electrons.

A wire moving in a magnetic field As the wire moves, the electrons also move from top to bottom. Thus the magnetic field will exert a force on these moving electrons (force on a moving charge, not current in a wire). The force on the electrons is directed from left to right and therefore the electrons are pushed to the right.

A wire moving in a magnetic field

Lastly It is no longer the case that the electric field inside a conductor is zero when charges are allowed to move (that was static electricity only).

Faraday’s Law As we saw earlier, an electric current creates a magnetic field. In the previous section we saw that a wire that moves in a magnetic field has an induced emf at its ends. Actually producing a current by a magnetic field was a difficult problem in nineteenth-century physics.

Faraday’s Law - Experiment Consider the following experiment. A magnet is moved towards a loop of wire whose ends are connected to a sensitive galvanometer and in a direction normal to the plane of the loop as shown. The galvanometer registers a current.

Faraday’s Law – Experiment observations If the magnet is simply placed near the coil, but does not move relative to it, nothing happens. The current has been created as a result of the motion of the magnet relative to the loop of wire. If we now move the magnet toward the coil faster, the reading on the galvanometer is greater. If we move the coil toward the magnet, we again find a reading.

Faraday’s Law – Experiment implications If we use a different loop of wire with the same area but more turns, the effect is greater. If the area of the single loop is increased, the current also increases. If the magnet is moved at an angle other than 90°, the current decreases. This indicates that it is the relative motion of the coil and magnet that is responsible for the effect. If a stronger magnet is used, the effect is greater.

Faraday’s law – Experiment summary The observations are that the current registered by the galvanometer increases when: The relative speed of the magnet with respect to the coil increases; The strength of the magnet increases; The number of turns increases; The area of the loop increases; The magnet moves at a right angle to the plane of the loop.

Faraday’s law – Magnetic Flux Faraday found that the common thread behind all these observations is the concept of magnetic flux. Imagine a loop of wire in a magnetic field whose magnitude and direction is constant, then we define flux as follows.

Faraday’s law – Magnetic Flux

This means that if the magnetic field is along the plane of the loop, then θ =90° and hence φ = 0. The maximum flux through the loop occurs when θ = 0°, when the magnetic field is parallel to the loop area and its value is then BA.

Faraday’s law – Magnetic Flux

Thus, to increase the magnetic flux of a loop of wire we must: Increase the loop area that is exposed to the magnetic field; Increase the value of the magnetic field; Have the loop normal to the magnetic field.

Example A loop of area 2 cm 2 is in a constant magnetic field of B = 0.10 T. What is the magnetic flux through the loop when: The loop is perpendicular to the field; The loop is parallel to the field; The normal to the loop and the field have an angle of 60° between them?

Faraday’s law - Magnetic Flux What does magnetic flux have to do with the problem of how a magnetic field can create an electric field? The answer lies in a changing magnetic flux. In all the cases we described, we had a magnetic flux through the loop, which was changing with time. As a magnet is brought closer to the loop area, the value of the magnetic field at the loop portion is increasing and so is flux. If the magnet is held stationary near the loop, there is flux through the loop, but it isn’t changing – so nothing happened. If the number of turns is increased, so is the flux linkage.

Faraday’s law - Magnetic Flux

Example The magnetic field through a single loop of area 0.2 m 2 is changing at a rate of 4 T/s. What is the induced emf?

Example A uniform magnetic field B = 0.40 T is established into the page. A rod of length L = 0.20 m is placed on a railing and pushed to the right with constant speed v = 0.60 m/s. What is the induced emf in the loop?

Lenz’s Law Thinking about the last example, we need to ask the question, which direction does the current flow? There are two options. If the current flows counter- clockwise, by the right hand rule, force is directed right – in the direction of motion of the rod. Or, if the current flows clockwise, the force is directed to the left – oposing te motion of the rod. Which makes physical sense?

Lenz’s Law If current flows counter-clockwise, then the rod is accelerated to the right. An increased speed leads to increased emf, which leads to increased current and so on and so on forever. But with no one providing the necessary energy, this scenario violates the conservation of energy.

Lenz’s Law Clearly, the induced current must act to oppose the change in magnetic flux that created the current. This is Lenz’s law.

Lenz’s Law This is a little tricky to think about, so lets stick with the scenario in the last example. The change in the magnetic flux was a decrease because the area got smaller. The induced current will create its own magnetic field (Ørsted’s discovery), which will also have a flux through the same loop.

Lenz’s Law If the created magnetic field is in the same direction as the original magnetic field, its flux will add to the original (which was decreasing) and prevent it from decreasing as fast. In this case, the induced current must flow in a clockwise direction, as we found earlier!

Example A loop of wire has its plane horizontal and a bar magnet is dropped from above so that it falls through the loop with (a) the north pole first and (b) the south pole first. Find the direction of the current induced in the loop in each case.

Summary State the formula to calculate Flux When is an emf induced? How can we calculate its magnitude? How can we determine the direction of the induced current in a wire?