A brief recap of the context…mcba11.phys.unsw.edu.au/~mcba/PHYS1231/FifthLecture.pdf · The...

27/08/2013

Today’s lecture:

PHYSICS 1B

Electricity & Magnetism

Motional emf

and

Lenz’s Law

A GFCI is a Ground Fault Circuit Interrupter. It is designed to protect users of electrical appliances against an electric shock. When the currents in the wires run in opposite directions, the flux is zero. When the return current in wire 2 changes, then the flux is no longer zero. The resulting induced emf can be used to trigger a circuit breaker.

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Applications of Faraday’s Law - GFCI

PHYSICS 1B – Faraday’s Law

Electricity & Magnetism – Induction & Inductance

The pickup coil of an electric guitar uses Faraday’s law. The coil is placed near the vibrating string and causes a portion of the string to become magnetised. When the string vibrates, the magnetised segment produces a changing flux through the coil. The induced emf is fed into an amplifier.

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Applications of Faraday’s Law – Pickup Coil



A motional emf is the emf induced in a conductor moving through a constant magnetic field. The conductor is in motion – hence the name! The electrons in the conductor experience a force:

𝑭𝑩 = 𝑞𝒗 × 𝑩 which is directed along 𝑙.

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Motional emf



Under the influence of the force, the electrons move to the lower end of the conductor and accumulate there. As a result of the charge separation, an electric field, E is produced inside the conductor. The charges accumulate at both ends of the conductor until they are in equilibrium with regard to the electric and magnetic forces.

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Motional emf



In equilibrium, then, the force due to the magnetic field is balanced by the force due to the electric field. i.e. 𝑞𝐸 = 𝑞𝑣𝐵 or 𝐸 = 𝑣𝐵 A potential difference is maintained between the ends of the conductor as long as it continues to move through the magnetic field. If the direction of the motion is reversed, then the sign of the potential difference will also be reversed.

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Motional emf



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Motional emf – a loop rotating in a magnetic field



The electrical resistance of an element of an electric circuit is the opposition to the passage of an electric current through that element.

An object of uniform cross section has a resistance proportional to its resistivity and its length, and inversely proportional to its cross-sectional area.

Almost all materials have some resistance, except for superconductors.

The resistance, R, of an object is defined as the ratio of the voltage across (V) it to the current through it (I). The unit of resistance is the Ohm (Ω).

i.e. 𝑅 =𝑉

𝐼

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Electrical Resistance



A bar moving through a uniform field is shown (to the left) and the equivalent circuit diagram is shown (to the right).

The wiggly line is a resistor, which has resistance R.

Assume that the bar has zero resistance.

The stationary part of the circuit has a resistance R. The work done by the applied force appears as internal energy in the resistor.

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The Sliding Conducting Bar



The magnetic flux is 𝛷𝐵 = 𝐵𝑙𝑥 The induced emf is therefore:

𝜀 = −𝑑𝛷𝐵𝑑𝑡

= −𝑑

𝑑𝑡𝐵𝑙𝑥 = −𝐵𝑙

𝑑𝑥

𝑑𝑡= −𝐵𝑙𝑣

Using our equation for resistance, the current is therefore:

𝐼 =𝜀

𝑅=𝐵𝑙𝑣

𝑅

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The Sliding Conducting Bar



The applied force, Fapp, does work on the conducting bar.

This moves the charges through a magnetic field.

The magnetic force, 𝐹𝐵 = 𝐵𝐼𝑙 opposes the motion Its direction is opposite to the applied force (check with the right hand rule!) Since the bar is moving at constant speed (i.e. we have no acceleration), we must have 𝐹𝑎𝑝𝑝 = 𝐹𝐵 = 𝐵𝐼𝑙

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The Sliding Conducting Bar: Forces



The change in energy must be equal to the transfer of energy into the system by the work done by the applied force on the conducting bar. So we’d have to model our circuit as a non-isolated system. The power input is equal to the rate at which energy is delivered to the resistor

Therefore: (since we said 𝐼 =𝐵𝑙𝑣

𝑅 )

𝑃 = 𝐹𝑎𝑝𝑝𝑣 = 𝐵𝐼𝑙 𝑣 =𝐵2𝑙2𝑣2

𝑅=𝜀2

𝑅

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The Sliding Conducting Bar: Energy Considerations



As an aircraft flies from Sydney to Melbourne, it passes through the Earth’s magnetic field, which is directed upwards. As a result, a motional emf is developed between the wingtips. Which wingtip is positively charged? a) The left wing

b) The right wing

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Quick Quiz



As an aircraft flies from Sydney to Melbourne, it passes through the Earth’s magnetic field, which is directed upwards. As a result, a motional emf is developed between the wingtips. Which wingtip is positively charged? b) The right wing

The Earth’s magnetic field has an upward component in the southern hemisphere. As the plane flies south, the right-hand rule indicates that positive charges will experience a force directed to the west. Thus, the right wingtip becomes positively charged, and the left wingtip becomes negatively charged.

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Quick Answer



In the figure, an applied force of magnitude Fapp results in a constant speed v and a power input P. Imagine that the force is increased so that the constant speed of the bar is doubled to 2v. Under these conditions, the new force and the new power input are: a) 2F and 2P

b) 4F and 2P

c) 2F and 4P

d) 4F and 4P

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Quick Quiz



Imagine that the force is increased so that the constant speed of the bar is doubled to 2v. Under these conditions, the new force and the new power input are: c) 2F and 4P

The force on the wire is of magnitude 𝐹𝑎𝑝𝑝 = 𝐹𝐵 = 𝐼𝑙𝐵,

with 𝐼 =𝐵𝑙𝑣

𝑅 .

Thus, the force is proportional to the speed and the force doubles. Because 𝑃 = 𝐹𝑎𝑝𝑝𝑣, the doubling of the force and the

speed results in the power becoming four times as large.

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Quick Answer



Faraday’s law indicates that the induced emf and the change in flux have opposite algebraic signs:

𝜀 = −𝑑𝛷𝐵𝑑𝑡

This has a physical interpretation that has come to be known as Lenz’s law. Developed by German physicist Heinrich Lenz (1804 – 1865). Also carried out research into the physical properties of seawater, as well as the climate around the world.

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Lenz’s Law

PHYSICS 1B – Lenz's Law

Electricity & Magnetism – Lenz's Law & Generators

Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop. In other words, the induced current tends to keep the original magnetic flux through the circuit from changing.

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Lenz’s Law



The conducting bar slides on the two fixed conducting rails. The magnetic flux due to the external magnetic field through the enclosed area increases with time as the bar is slid to the right.

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Lenz’s Law, Example



Since the flux through the enclosed area is increasing, the induced current must produce a magnetic field out of the screen. From the right hand rule, we therefore see that the induced current must be counter-clockwise.

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If we were to move the bar in the opposite direction (panel b), then the flux through the loop decreases with time. The direction of the induced current would therefore be clockwise, producing a magnetic field into the screen.

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A magnet is placed near a metal loop. If the magnet is pushed towards the loop (panel a), the induced current produces a field directed to the left to counteract the increasing external flux (panel b). If the magnet is moved away from the loop (panel c), then the induced current produces a field directed to the right to counteract the decreasing external flux (panel d)

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Induced Current Directions: Example



In the figure, a magnet is being moved in the vicinity of a solenoid that is connected to a very sensitive ammeter. The south pole of the magnet is nearest the solenoid, and the ammeter indicates a clockwise current (as viewed from above) in the solenoid. The person is: a) Inserting the magnet

b) Pulling it out

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Quick Quiz



The person is: a) Inserting the magnet Because the current induced in the solenoid is clockwise, as viewed from above, the magnetic field lines produced by the current point downwards in the figure. Thus, the upper end of the solenoid must be acting as a magnetic south pole. For this situation to be consistent with Lenz’s law, the south pole of the bar magnet must be approaching the solenoid. Remember: like poles repel!

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Quick Answer



In this figure, a circular loop of wire is being dropped towards a wire that carries a current that is flowing to the left. The direction of the induced current in the loop of wire is: a) Clockwise

b) Counter-clockwise

c) Zero

d) Impossible to determine

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Quick Quiz



The direction of the induced current in the loop of wire is:

b) Counter-clockwise At the position of the loop, the magnetic field lines point into the page. The loop is entering a region of stronger magnetic field as it drops towards the wire, so the flux is increasing. The induced current must set up a magnetic field that opposes this increase. To do this, it creates a magnetic field directed out of the page. This requires a counter-clockwise current in the loop (from the right-hand rule). 27/08/2013

Quick Answer



An electric field is created in the conductor as a result of the changing magnetic flux. Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space. The induced field is non-conservative, unlike the electric field produced by stationary charges.

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Induced emf and Electric Fields



The emf, ε, for any closed path can be expressed as the line integral of E.ds over that path - i.e.

𝜀 = 𝑬. 𝑑𝒔

Therefore, we can write Faraday’s law in a general form:

𝑬. 𝑑𝒔 = −𝑑𝛷𝐵𝑑𝑡

The field cannot be an electrostatic field (i.e. one created by a charge), since electrostatic fields are conservative, and therefore the integral around a closed path would be zero. It isn’t! 27/08/2013

Induced emf and Electric Fields



In a region of space, the magnetic field increases at a constant rate. This changing magnetic field induces an electric field that: a) Increases in time

b) Is conservative

c) Is in the direction of the magnetic field

d) Has a constant magnitude

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Quick Quiz



In a region of space, the magnetic field increases at a constant rate. This changing magnetic field induces an electric field that: d) Has a constant magnitude

The constant rate of change of B will result in a constant rate of change of the magnetic flux.

According to our general version of Faraday’s law, 𝑬. 𝑑𝒔 = −𝑑𝛷𝐵

𝑑𝑡 , if

𝑑𝛷𝐵

𝑑𝑡 is

constant, then E is constant in magnitude.

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Quick Answer



A brief recap of the context…mcba11.phys.unsw.edu.au/~mcba/PHYS1231/FifthLecture.pdf · The...

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