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**Faraday’s law of induction**

Chapter 31: Faraday’s law of induction Reading assignment: Chapter 31 Homework 31.1 (due Wednesday, Nov. 19): OQ1, OQ2, OQ3, OQ4, OQ5, QQ1, QQ3, 1, 3, 4, 5, 6, 22 Homework 31.2 (due Monday, Nov. 24): OQ6, OQ7, OQ9, OQ10, OQ11, 27, 29, 42, 43, 53, 54, 63 Previous chapter: A current produces a magnetic field Now: Can a magnetic field produce an electric current? Yes – Faraday’s law of induction: A changing magnetic field (in time) induces an emf (and, thus, also a current) An emf (and therefore a current as well) can be induced in various processes that involve a change in magnetic flux. Faradays law of induction: This is the underlying physical law for generators and electric power generation extremely powerful and useful applications (electric power generation).

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**Faraday’s law of induction:**

Can a magnetic field produce an electric current? Faraday: A steady magnetic field produces no current Faraday: A changing magnetic field does produce a current. This current is called induced current. Induced emf is produced by a changing magnetic field Negative current No current Positive current

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**Remember Magnetic flux**

Product of the normal component of the magnetic field passing through a loop of area A, times area A. Unit of magnetic flux FB is Weber (1W) = Tesla·meter2 The strength of the B field is proportional to the number of lines per unit area. Thus the flux FB is proportional to the total number of normal lines passing through the coil.

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**Faraday’s law of Induction**

If the flux through N loops of wire changes by an amount DFB during a time Dt, the induced emf during this time is: An emf can be induced in three ways (and any combination of those): By changing the magnitude of the magnetic field, B. By changing the area of the loop, A. By changing the loops orientation with respect to the field, q.

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White board examples A coil consists of 2000 turns of wire. Each turn is a square of side d = 18 cm, and a uniform magnetic field directed perpendicularly to the plane of the coil is turned on. If the field changes from 0 to 0.5 T in 0.8 s, what is the magnitude of the induced emf in the coil, while the field is changing? The flexible loop in the figure below has a radius of 12.0 cm and is in a magnetic field of magnitude T. The loop is grasped at points A and B and stretched until its area is nearly zero. If it takes s to close the loop, what is the magnitude of the average induced emf in it during this time interval?

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**Application – induction stove**

Some modern stove burners a based on induction. An ac current passes around a coil that is the “burner” (that never gets hot). Why will it heat a metal pan, but not a glass pan? The AC current sets up a changing magnetic field that passes through the bottom of the pan. This changing magnetic field induces a current through the pan bottom, and since the pan offers resistance, electric energy is transformed into heat, heating the pot and its contents. Works very well with steel pots. Does not work well with aluminum pots (magnetic field does not penetrate far enough no good induction current). Does also not work on non-conducting pots. Resistance in glass container is too high, very little current is induced.

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**Application - Ground Fault Circuit Interrupters**

Fuses/circuit breakers don’t keep you from getting electrocuted But GFI’s (or GFCI’s) do GFCI Under normal use, the current on the live wire matches the current on the neutral wire. Magnetic fields cancel inside orange donut. Now, imagine you touch the live wire – current path changes (no current goes back to ground wire). There is magnetic field around the donut Changing magnetic field means EMF in blue wire Current flows in blue wire Magnetic field produced by solenoid Switch is magnetically turned off

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**Lenz’s law – what is the direction of the current?**

An induced emf always gives rise to a current whose magnetic field opposes the original change in flux Example:

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**Examples Practice with Lenz’s law.**

In which direction is the current induced in the coil for each situation in the Figure? Counterclockwise current Clockwise current No current Can’t be determined

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**Examples Practice with Lenz’s law. Counterclockwise current**

In which direction is the current induced in the coil for each situation in the Figure? Counterclockwise current Clockwise current No current Can’t be determined

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**Black board example Pulling a coil from a magnetic field.**

A square coil of 5.0 cm contains 100 loops and is positioned perpendicular to a uniform 0.60 T magnetic field. It is quickly and uniformly pulled from the field (perpendicular to B) to a region where B drops abruptly to zero. It takes 0.1 seconds for the coil to reach the field-free region. (a) Find the change in flux through the coil. (b) Find the emf and current induced if the resistance of the coil is 100 W. How much energy is dissipated in the coil if its resistance is 100 W?

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**Motional emf - emf induced in a moving conductor**

Look at the Figure: Uniform magnetic field is perpendicular to the area bounded by the U-shaped conductor. Movable rod resting on it. Rod is moving at speed v; travels a distance Dx = v·Dt Area of the loop increases DA = l·Dx = l·v·Dt By Faraday’s law there is an induced emf: Direction of current can be obtained by right hand rule (remember F = qvB) or Lenz’s law.

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**White board example Pulling a coil through a magnetic field.**

A rectangular metallic loop of dimensions l and w and resistance R moves with constant speed v to the right as shown in the figure. The loop passes through a uniform magnetic field directed into the page and extending 3w along the x-axis. Plot the magnetic flux, the emf and the applied force.

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**Faraday’s law in general form: A changing magnetic flux produces an electric field**

We just saw that a changing magnetic flux induces an emf, and a current in a conducting loop. A current implies there is an electric field in the conductor. Thus, a changing magnetic field (changing magnetic flux) induces an electric field! The E-field is perpendicular to the B-field

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**(here: mostly AC generators)**

Electric Generators (here: mostly AC generators) Or: How to convert mechanical into electrical energy? Many coils of wire (only one is shown) rotate in a magnetic field. The axle is turned by some mechanical means. An emf is induced in the rotating wire and electric current is generated. The generated current is alternating current. Right hand rule for induction: thumb – Induced current (Force on charge) index finger – velocity of wire; middle finger – B-field v v

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**𝛷 𝐵 = 𝐵 ∙ 𝐴 =𝐵⋅𝐴∙ cos 𝜃 𝛷 𝐵 =𝐵⋅𝐴∙ cos 𝜔𝑡 Electric power generation:**

emf (voltage) produced by a generator Magnetic flux: 𝛷 𝐵 = 𝐵 ∙ 𝐴 =𝐵⋅𝐴∙ cos 𝜃 𝛷 𝐵 =𝐵⋅𝐴∙ cos 𝜔𝑡 The coil is rotating with angular velocity w. Then q = wt For N loops, we get an emf of: v

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**White board problem loop of wire**

A rectangular loop of wire 20 cm by 20 cm with 50 turns is rotated rapidly in a magnetic field B, so that the loop makes 60 full rotations a second. At t = 0 the loop is perpendicular to B. What is the EMF generated by the loop, in terms of B at time t? What B-field do we need to get a maximum voltage of 170 V? loop of wire

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**Comments on Generators**

The EMF generated is sinusoidal in nature (with simple designs) This is called alternating current - it is simple to produce This is actually how power is generated Generators extremely similar to motors – often you can use a single one for both Turn the axle – power is generated Feed power in – the axle turns Regenerative braking for electric or hybrid cars

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Induced Voltages and Inductance

Induced Voltages and Inductance

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