PHYSICS - GIANCOLI CALC 4E CH 29: ELECTROMAGNETIC INDUCTION.

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2 CONCEPT: ELECTROMAGNETIC INDUCTION A coil of wire with a VOLTAGE across each end will have a current in it - Wire doesn t HAVE to have voltage source, voltage can be INDUCED i V Common ways to INDUCE a voltage on a coil of wire: - Move a bar magnet - Vary current in electromagnet - Turn electromagnet on and off Induced voltage known as INDUCED EMF INDUCED CURRENT - Process known as ELECTROMAGNETIC INDUCTION EXAMPLE of current induction: vሬԧ vሬԧ = 0 vሬԧ CURRENT INDUCED NO CURRENT INDUCED CURRENT INDUCED CERTAIN changes will induce a current, and the magnitude of the current depends on the rate of these changes - ar magnet moving into coil Faster it goes, larger the induced current - Current changing in electromagnet near a coil Faster the current changes, larger the induced current Page 2

3 CONCEPT: MAGNETIC FLUX Remember! Electric flux is just the amount of passing through a surface - MAGNETIC FLUX is just the amount of passing through a surface MAGNETIC FLUX is given by A θ Normal Φ = (UNITS 1 Wb = 1 ) Magnetic flux changes with,, and - IMPORTANT to remember this changes in magnetic flux will be important later! EXAMPLE: What is the magnetic flux through the square surface depicted in the following figure, if = 0.05 T? Assume the side length of the square is 5 cm. Surface 30 o Page 3

4 PRACTICE: MAGNETIC FLUX THROUGH A RING A ring of radius 0.5m lies in the xy-plane. If a magnetic field of magnitude 2 T points at an angle of 22 o above the x-axis, what is the magnetic flux through the ring? EXAMPLE: ROTATING RING A ring of radius 2 cm is in the presence of a 0.6 T magnetic field. If the ring begins with its plane parallel to the magnetic field, and ends with the plane of the ring perpendicular to the magnetic field, what is the change in the magnetic flux? Page 4

5 CONCEPT: MAGNETIC FLUX WITH CALCULUS Magnetic flux when the magnetic field DOESN T CHANGE over the surface Φ = Acosθ - For changing fields Φ = EXAMPLE: What is the flux through the loop shown in the following figure? i d w L Page 5

6 CONCEPT: FARADAY S LAW Changing magnetic field through conducting loops - This is actually due to a changing MAGNETIC FLUX A changing MAGNETIC FLUX leads to an induced EMF: Ɛ ind = - This is known as Faraday s Law Remember! Φ = A cos θ - So, magnetic flux changes with,, and EXAMPLE 1: A square conducting wire of side length 4 cm is in a 2 T magnetic field. It rotates such that the angle of the magnetic field to the normal of the square increases from 30 o to 60 o in 2 s. What is the induced current on the wire if its resistance is 5 Ω? Page 6

7 PRACTICE: FARADAY S LAW AND TWO SOLENOIDS Two solenoids are placed end to end, with one solenoid connected to a variable power source, and the other solenoid connected to a 10 Ω resistor. The first solenoid has 10 turns per cm and has as an initial current of 2 A, and the second solenoid has 5 turns and a radius of 2 cm. a) What is the change in magnetic field emitted by the first solenoid if the current increases from 2 A to 5 A in 1 s? b) What is the change in the magnetic flux through the other solenoid during this 1 s? c) What is the induced EMF on the second solenoid? d) What is the induced current on the second solenoid? EXAMPLE: CURRENT IN A CIRCUIT WITH A CHANGING, EXTERNAL MAGNETIC FIELD What current does the ammeter read if the following circuit, with an area of 50 cm 2, is placed in a magnetic field that is changing at 0.05 T/s? Note that the resistor has a resistance of 2 Ω. ሬሬԦ A Page 7

8 CONCEPT: FARADAY S LAW WITH CALCULUS FARADAY S LAW states, more precisely, - Ɛ = EXAMPLE: Find the current in the loop shown in the following figure if the current in the straight wire increases at di/dt. i d w L b Reminder! etween any two points, a and b, V ab = E dl a - For a coil which has an induced EMF, the integral has to be a CLOSED integral over the loop E dl = dφ dt Page 8

9 EXAMPLE: RAIL GUN A rail gun is composed of a moveable conducting rod on a U-shaped circuit, as shown below. What is the next force on the conducting rod in this case? Page 9

10 CONCEPT: LENZ S LAW Faraday s Law tells us the magnitude of the induced EMF magnitude of induced current - To find DIRECTION of induced current, we use Lenz s Law LENZ s LAW states: A conductor will induce a magnetic field on itself to changes in its magnetic flux v v ind v ind v Once the direction of the induced magnetic field is known, right hand rule gives direction of induced current v ind v i ind EXAMPLE: In the following scenarios, find the direction of the current induced on the conductors. v v v Page 10

11 PRACTICE: DIRECTION OF INDUCED CURRENT IN A RING What is the direction of the induced current in the inner ring shown in the following figure? For this problem, consider the battery s voltage as continuously INCREASING. Note: the arrow striking through the battery in the circuit diagram indicates that the voltage of the battery is variable (i.e. it can be changed). EXAMPLE: AR MAGNET VS CURRENT-CARRYING WIRE A bar magnet moves relative to a coil of wire as indicated in the figure below and induces a current in the coil. A current carrying wire carries a current relative to a coil as shown in the second figure. Would you need to increase or decrease the magnitude of the current in the wire to induce a current in the coil that moves in the SAME direction as the current induced by the bar magnet? Scenario 1 v Scenario 2 i Page 11

12 CONCEPT: MOTIONAL EMF Remember! A changing magnetic field can produce an EMF - UT so can motion. This is referred to as a MOTIONAL EMF. If a conductor moves through a magnetic field, charges feel a Positive charges feel the force [ UPWARD / DOWNWARD] Separation of charges L FሬԦ ሬሬԦ vሬԧ Separation of charges E field Electric force that magnetic force - To balance, E = v Induced EMF Ɛ = EL = EXAMPLE: If a conductor of length 10 cm moves with a velocity of 20 m/s in a magnetic field of 0.05 T, what is the current through the conductor if its resistance is 15 Ω If a conductor moves along U-shaped wire, MAGNETIC FLUX changes - Change in Change in magnetic flux Production of L ሬሬԦ vሬԧ Change in area of Change in magnetic flux of Induced EMF Ɛ = ΔΦ Δt = EXAMPLE: In the circuit below, if the wire has a resistance of 10 Ω, what is the current induced if the length of the bar is 10 cm, the speed of the bar is 25 cm/s, and the magnetic field is 0.02 T? What about the power generated by the circuit? a ሬሬԦ vሬԧ b Page 12

13 PRACTICE: AR MOVING IN UNKNOWN MAGNETIC FIELD A thin rod moves in a perpendicular, unknown magnetic field. If the length of the rod is 10 cm and the induced EMF is 1 V when it moves at 5 m/s, what is the magnitude of the magnetic field? Page 13

14 CONCEPT: TRANSFORMERS Power in North America is delivered to outlets in homes at 120 V. - This is too large to operate many delicate electronics, such as computers. Remember! A coil with a changing magnetic field can induce an EMF on a second coil - This induced EMF can be as small as needed. A TRANSFORMER does exactly this it uses Faraday s law to convert a large voltage to a small EMF: V 1 V 2 The ratio of the VOLTAGES in a transformer depends upon the ratio of the TURNS: V 2 V 1 = N 2 N 1 EXAMPLE: You need to build a transformer that drops the 120 V of a regular North American outlet to a much safer 15 V. You already have a solenoid with 50 turns made, but you need to make a second solenoid to complete your transformer. What is the least number of turns the second solenoid could have? Page 14

15 PRACTICE: OPERATING A LAPTOP An outlet in North America outputs electricity at 120 V, but a typical laptop needs to operate at around 20 V. In order to do so, a transformer is placed in a laptop s power supply. If the coil in the circuit connected to the laptop has 20 turns, how many turns must the coil in the circuit with the outlet have? Page 15