AP Physics C - E & M

AP Physics C - E & M Electromagnetic Induction 2017-07-14 www.njctl.org

Table of Contents: Electromagnetic Induction Click on the topic to go to that section. Induced EMF Magnetic Flux and Gauss's Law Faraday's Law of Induction EMF induced in a moving conductor Lenz's Law Maxwell's Equations Move any photo or image in this presentation to reveal a link to its source, providing attribution and additional information.

Induced EMF (Electromotive Force) Return to Table of Contents

Electromotive Force (EMF) Electromotive Force is actually a potential difference between two points that is measured in Volts. It is NOT a force, but it is an historical term that has not gone away. Because it is an unfortunate name, it is frequently just referred to as EMF or. In the Circuits unit, it represented the maximum voltage developed by a battery. This unit will show a way that a voltage and a current can be developed in a conducting wire that is not connected to a battery. When these two quantities are not caused by a battery, they are called induced EMF and induced current.

Induced EMF Oersted, and then, Ampere showed that a current will generate a magnetic field which will interact with other currents and magnetic materials. This led to other physicists experimenting to see if this would work in reverse - could a magnetic field generate a current? Michael Faraday was able to make this connection in 1831 - but it was slightly different. Unlike a steady current generating a magnetic field, a steady magnetic field would not generate a current.

Induced EMF Either the Magnetic Field, or the cross sectional area of the conductor through which the charge would flow, had to change to produce a current. The Magnetic Field strength could increase or decrease, or the angle at which it passed through the cross sectional area could change. In America, Joseph Henry performed a similar experiment at the same time, but did not publish it - hence Faraday's name is more closely associated with this effect. This happens a lot in Mathematics and Physics - Newton (in the U.K.) and Leibniz (in Germany) developed related forms of Calculus at the same time, independent of each other. But, Joseph Henry had the unit of inductance named after him. Faraday has a constant named after him - the magnitude of charge of a mole of electrons, and a unit - the Farad, for capacitance.

Induced EMF Michael Faraday connected a battery to a metal coil via insulated wires (the coil increased the magnetic field) and found that a current would be induced in the current loop on the right when the switch on the left side was closed and opened. There is zero current present in the coil at all times. These are insulated wires, and any current present in them is NOT passing through the metal coil.

Induced EMF When the current supplied by the battery is steady, the current in the loop on the right side is zero. When the switch is initially closed, current will flow in the secondary loop until the primary current reaches its constant value. Primary Secondary When the switch is then opened, current will momentarily flow again - but in the opposite direction!

Induced EMF Faraday's Disk Generator - by spinning the metal disk between the poles of the U shaped magnet (A), the changing magnetic field will induce an EMF, and hence, a current in the disk (D), which will flow out of the machine via terminals B and B'. A bar magnet that moves towards or away from a loop of wire will generate an EMF, and then a current in the loop.

Induced EMF This now provided evidence that a magnetic field could generate a current. But, there is a difference. A steady current will generate a magnetic field. But, a steady magnetic field and a non moving, constant area loop of wire will NOT result in a current in the wire. A constant magnetic field and a moving loop of wire will result in a current. A changing (magnitude or orientation) magnetic field and a stationary loop of wire will result in a current. A constant magnetic field and a changing area of the loop of wire will result in a current. We need to define Magnetic Flux before we can fully understand this phenomenon.

1 A bar magnet is moved towards a circular conducting loop. As this occurs: A The magnetic field in the loop does not change, and no current flows in the loop. B The magnetic field in the loop decreases and no current flows in the loop. C The magnetic field in the loop decreases and a current flows in the loop. Answer D The magnetic field in the loop increases and a current flows in the loop. E The magnetic field in the loop increases and no current flows in the loop.

2 Which of the following cases will generate an induced EMF? A I, II and III B I and II C I and III D II and III E III only I - An external magnetic field increases its magnitude through the loop. II - An external magnetic field changes its orientation to the loop's cross sectional area. III - A magnet moves with a velocity, v in the +x direction towards a loop, which also moves with velocity, v, in the +x direction. Answer

3 In which of the following diagrams will a current not be induced in the conducting loop? A B C D Answer E

Magnetic Flux and Gauss's Law Return to Table of Contents

Magnetic Flux Magnetic Flux describes the quantity of Magnetic Field lines that pass in a perpendicular direction through a given surface area and is represented by: θ is the angle between the Magnetic Field and the normal vector to the surface area. The unit of Magnetic Flux is the Weber, Wb, where 1 Wb = 1Tm 2. It is calculated in the same method as we did earlier with the Electric Flux - and the next two slides summarize the work - but this time using a Magnetic Field and not Electric Field.

Magnetic Flux Even though the Electric Field lines are quite different from Magnetic field lines, the calculation of the flux is the same. Electric Field lines start on a charge and never come back to the charge. Magnetic Field lines start on a North Pole, and return to the South Pole of the same origin - whether it's a bar magnet or an atom. This will have an effect when we show Gauss's Law for Magnetism - but for now, let's review the flux calculation.

Magnetic Flux If the Magnetic Field is not perpendicular to the area of space that we're interested in finding out the Magnetic Flux, such as the area, A, below, then we need the concept of a normal vector, da. A is the area of the slanted side and A 1 is the area of the vertical side. The Magnetic Field lines make an angle θ with the vector da which is normal (perpendicular) to the light blue surface area.

The number of field lines through surfaces A 1 and A is the same. The widths of each surface are the same, and their lengths are two sides of a triangle where A has the length of the hypotenuse. Magnetic Flux Since the flux through both surfaces is the same: The flux is dependent on the number of field lines, and proportional to the angle that the normal to the surface makes with the field.

Magnetic Flux Find the magnetic flux through the rectangular loop adjacent to the current carrying wire below. l w a The magnetic field inside the loop is perpendicular to da, the normal vector to the loop and is directed out of the page and at any point, y, where y is the distance from the wire, is: I

Magnetic Flux l w a y dy da The incremental magnetic flux within the loop is: I B is perpendicular to da da is the light blue shaded area illustrated above and is equal to ldy.

Magnetic Flux l w a y dy da Integrate over dy from a to a+w, to find the total flux: I

Gauss's Law First, a review of Gauss's Law for an Electric Field. A Gaussian surface is drawn about charge q. The Electric Field lines emanate from the charge and go off to terminate on a negative charge or infinity. The flux through the closed surface is proportional to the enclosed charge - Gauss's Law.

N S Gauss's Law Draw a closed Gaussian surface around the magnet to the left (again, this can be a bar magnet or an atom - which has its own magnetic field due to the electron motion). Count the amount of field lines entering and subtract the number of lines leaving - the same convention we used to get the sign of Electric Field lines entering and leaving a surface. What do you get?

Gauss's Law N S You get zero. For a closed surface about a magnetic field source, there is zero flux through it. The only way this worked for an Electric Field is if the surface did not include a charge. It works all the time for the Magnetic Field. Gauss's Law for Magnetism:

Faraday's Law of Induction Return to Table of Contents

Faraday s Law of Induction Michael Faraday and Joseph Henry showed that a changing current will induce an EMF which creates an electric current in a second loop. Their initial experiments showed that a changing current generates a changing magnetic field which develops an EMF and current. What is actually changing is the Magnetic Flux. Faraday's Law of Induction states that the induced EMF in a wire loop is proportional to the rate of change of Magnetic Flux through the loop: If there are N loops, then the induced EMF is:

Faraday s Law of Induction For now, we'll not talk about the negative sign - that is explained by Lenz's Law, which is in a couple of chapters. Just work with the magnitude. By changing the magnitude of the Magnetic Field or cross sectional area of the conductor or the angle that the field makes with the normal to the cross sectional area, an EMF will be generated in the conductor. If the angle and the cross sectional area remains constant, then the equation is: Similar equations are used for combinations where the different parameters change with time.

4 A wire loop of area A is placed in a time varying but spatially uniform magnetic field that is perpendicular to the plane of the loop. The induced EMF in the loop is, where b is a constant. The magnitude of the time varying magnetic field is: A B C Answer D E

5 A 300 loop coil of wire of radius 6.2 cm is placed in a Magnetic Field that increases from 0 to 0.65 T in 1.1 s, and is perpendicular to the plane of the coil. What is the EMF generated while the Magnetic Field is changing? A.0071 V B.0078 V C 1.8 V Answer D 2.1 V E 2.4 V

6 A 300 loop coil of wire of radius 6.2 cm is placed in a steady Magnetic Field of 0.65 T that is perpendicular to the coil's plane. The coil rotates 30 0 in 0.54 s. What is the EMF generated? A.009 V B.013 V C 0.58 V Answer D 2.0 V E 3.8 V

7 A loop of wire is placed in a constant Magnetic Field and begins to rotate about its diameter. Which of the graphs represents the magnetic flux through the loop as a function of time? Φ Φ A B t t Answer C Φ D Φ E Φ t t t

8 A circular loop of wire with radius, a, is placed in a uniform Magnetic Field perpendicular to the plane of the loop. The field varies with time, generating an EMF of where is a constant. Assuming B(t =0) = 0, which of the following represents the Magnetic Field as a function of time? A B C Answer D E

9 A circular loop of wire with radius, a, is placed in a Magnetic Field perpendicular to the plane of the loop, that varies with time as where is a constant. Which of the following represents the magnitude of the EMF as a function of time? A B Answer C D E

EMF induced in a moving conductor Return to Table of Contents

EMF induced in a moving conductor A changing Magnetic Flux occurs when the Magnetic Field changes, either in orientation or magnitude within a conducting loop. This is pretty straightforward - the angle can be changed by rotating the loop and the magnitude is changed by adjusting the field strength. The third method - changing the area of the loop - is not as obvious. It involves moving a conductor through a constant Magnetic Field. Let's start with what happens to this conductor.

EMF induced in a moving conductor + An external force (a push rod or the like) pushes a conducting rod of length l to the right in a Magnetic field that is directed into the page. - v The electrons in the rod feel a force, which moves them to the bottom of the rod. There is now a separation of charge, which creates an Electric Field directed from the top of the rod to the bottom, exerting an electric force on the electrons in the upwards direction.

EMF induced in a moving conductor + The electrons will continue moving until the magnitude of the two forces are balanced - Newton's Second Law again. v It is a uniform Electric Field, so: - Substituting, we find the potential difference between the top and bottom of the rod.

EMF induced in a moving conductor + v Add two parallel conducting rails for the rod to slide on and connect them to a resistor. We've created a circuit, where the source of the EMF is the moving rod. As long as the rod is moving, current will flow in a counter - clockwise direction. -

EMF induced in a moving conductor + We get the same result, invoking Faraday's Law, where the area of the loop at any time is the length of the rod, l, and the width of the circuit loop, x. v - Again, we won't worry about the sign, so the current due to this induced EMF is:

EMF induced in a moving conductor + - v The direction of the current was determined by balancing the electric and magnetic forces on the free charges in the rod. The ΔV resulting from the separation of the charges caused current to flow in a counter - clockwise direction. Lenz's Law will also explain the current direction - coming up in the next chapter.

EMF induced in a moving conductor + v Once current starts flowing, the external Magnetic Field will create a force on the rod that opposes the direction of the applied force. It will get harder to push the rod. - Use the direction of the current in the rod to obtain the correct direction for the magnetic force.

10 A circular wire loop travels with a velocity, v, through a Magnetic Field. Which of the following will not induce a current in the loop? A Increasing the Magnetic Field strength. B Expanding the loop, making a larger circle. C Moving the loop parallel to the Magnetic Field. Answer D Removing the loop from the field. E Rotating the loop about its diameter.

11 A square wire loop with side length, s, is placed in a perpendicular Magnetic Field of strength B. Which of the following represents the magnetic flux through the loop? A Blv B Bs 2 C 0 Answer D Bv/s 2 E B 2 v

Lenz's Law Return to Table of Contents

Lenz s Law Now it's time to explain the minus sign in Faraday's Law. It's so important that it has its own law! The minus sign tells us that the direction of the induced EMF in a current loop is such that the resulting current produces a magnetic field that resists the change of flux through the loop. This is a direct result of the Law of the Conservation of Energy. If the external field gets weaker, the induced current tries to replace the "missing" external field. If the external field gets stronger, the induced current opposes the "extra" external field. Only the Magnetic Field within the loop counts ; disregard the Magnetic Field outside.

Initial External Field (red).................... Final External Field Start with a magnetic field out of the page that decreases to zero. Lenz s Law Field due to Induced Current (blue).................... The changing magnetic field will induce an EMF in the loop that will generate a current in the counterclockwise direction. This induced current creates a field out of the page to oppose the decrease in the external field.

Initial External Field (red) Final External Field (red).................... Start with no magnetic field that increases to a magnetic field out of the page. Lenz s Law Field due to Induced Current (blue) and External.... x x x x Field. x. x x.. x. x x. x.. x. x. x x. x. x x x x.... The changing magnetic field will induce an EMF in the loop that will generate a current in the clockwise direction. This current creates a field into the page to oppose the increase in the external field.

Lenz s Law There are many other situations that can be analyzed with Lenz's Law, by using the following instructions. The Magnetic Field due to the induced current: 1. Points in the opposite direction to the external Magnetic Field if the external Magnetic Flux is increasing. 2. Points in the original direction of the external Magnetic Field if it is decreasing. 3. Is zero if the flux is not changing (it is zero because of Faraday's Law - there is no induced EMF if the Magnetic Flux is constant). Remember that the external Magnetic Field and the Magnetic Field due to the induced current are different.

Lenz s Law The direction of the induced current in a conducting loop is found to be the same whether using Lenz's Law or the analysis used in the Induced EMF chapter that found ΔV due to the charge separation in the conducting rod as it moved through a Magnetic Field. That's good. How is Lenz's Law consistent with the Conservation of Energy? We'll apply Proof by Contradiction to the moving conducting rod in a Magnetic Field.

Lenz s Law + - v As the rod moves to the right, the magnetic flux through the loop created by the rod and the rails increases. Lenz's Law states that the induced current must generate a Magnetic Field that opposes the increased flux - hence, the current flows counter - clockwise. The same result was obtained using the ΔV calculation.

Lenz s Law + - v What if we said the induced current was in the clockwise direction? The current would move from the top of the rod to the bottom. By, the magnetic force on the rod would be to the right - supporting, not opposing the applied force. The rod will accelerate.

Lenz s Law + The rod's velocity will increase, increasing the area of the loop, and the flux, which will increase the induced current. v This will increase the magnetic force on the rod, which will increase its velocity further. - Just by pushing the rod a little at the beginning will cause its velocity to increase without limit. This does NOT happen, and it also violates the Conservation of Energy.

Have you ever noticed that when you unplug an appliance that is running, there is a spark that jumps between the wall socket and the plug? This is explained by Lenz's Law. Lenz s Law As the plug is pulled out, the current decreases, collapsing its Magnetic Field. The change in magnetic field induces an EMF which produces a current which is seen as a spark. This is one reason why you should always turn off appliances before you unplug them. The energy that was stored in the magnetic field transformed into the electrical energy of the spark.

12 A magnetic field is pointing straight up through a coil of wire. The field is switched off. What is the direction of the induced current in the wire loop? A Out of the page. B Into the page. C Clockwise. D Counter-clockwise. E There is no induced current. Answer

13 A magnetic field is pointing straight up through a coil of wire. The field is doubled in magnitude. What is the direction of the induced current in the wire loop? A Out of the page. B Into the page. C Clockwise. D Counter-clockwise. E There is no induced current. Answer

14 A coil of wire is sitting on a table top. A magnet is held above it with its North Pole pointing downwards. What is the direction of the induced current in the coil of wire? A Out of the page. B Into the page. C Clockwise. D Counter-clockwise. E There is no induced current. Answer

15 A coil of wire is sitting on a table top. A magnet is held above it with its North Pole pointing downwards and is then pushed down towards the coil. What is the direction of the induced current in the coil of wire? A Out of the page. B Into the page. C Clockwise. D Counter-clockwise. E There is no induced current. Answer

Faraday's Law (complete) We are now in a position to complete Faraday's Law. E Faraday's Law and Lenz's Law show that an electric current is generated in the loop in a clockwise direction if the Magnetic Field out of the page increases. This implies an Electric Field in the same direction as the current, as the force that drives the positive charges is due to an Electric Force.

Faraday's Law (complete) The Electric Force due to the field is: E Calculate the work done by this force as a charge moves once around the loop - a line integral. Also, from the definition of Electric Potential Energy, the Work Energy Equation and the induced EMF generating a potential difference:

Faraday's Law (complete) E The Work calculated by both methods must be the same. Set them equal and use Faraday's Law to find the induced Electric Field due to a changing Magnetic Field.

Faraday's Law (complete) Earlier, the Electric Field was defined as a property of space surrounding a charge - it did not require any other charges. Extending that logic, if an EMF is induced by a changing Magnetic Field, it doesn't even need a conductor, as the charges are not needed. The Electric Field is generated independent of the presence of charge. For any geometry, the EMF is represented by the line integral of the Electric Field over a closed path - a line integral: Substitute this into the version of Faraday's Law that we've been using:

Faraday's Law (complete) Unlike the electrostatic Electric Field generated by a charge, the Electric Field generated by a changing Magnetic Field is a non-conservative field. The line integral over a closed path is NOT zero. Hence, it is a non-conservative field. Contrast with the line integral of an electrostatic Electric Field over a closed path: But don't confuse it with the surface integral of an Electric Field over a closed surface.

Maxwell's Equations Return to Table of Contents

Maxwell's Equations Maxwell's equations are four equations that show the relationship between electric fields and magnetic fields. This is not all Maxwell's work, but like Newton and his laws, he is credited for discovering the relationships, and providing a mathematical formalism. The integral forms of Maxwell's Equations are presented here. There are many ways to represent the equations - another common method is expressing them in terms of vector calculus - the del operator, cross products and dot products. Maxwell added the displacement current to Ampere's Law. It is a current between the plates of a charging capacitor - there is no movement of charges, but there is a changing Electric Field - a different notion of "current."

Maxwell's Equations Gauss's Law for an Electric Field: A charge generates an Electric Field. Gauss's Law for a Magnetic Field: Magnetic monopoles do not exist. Ampere's Law with the displacement current: Magnetic fields are generated by a current or/and a changing electric field. Faraday's Law: An electric field is generated by a changing magnetic flux.

16 Which of the following equations states: "Magnetic monopoles do not exist?" A B C D Answer E

17 Which of the following equations states: "A Magnetic Field is generated by a current?" A B C Answer D E

18 Which of the following equations includes Maxwell's contribution of the displacement current? A B C Answer D E

19 Which of the following equations states: "A changing Magnetic field generates an induced EMF?" A B C Answer D E

20 Which of the following equations states: "An electric charge generates an Electric Field?" A B C D Answer E