Right-hand Rule 2 gives direction of Force on a moving positive charge Right-Hand Rule Right-hand Rule 1 gives direction of Magnetic Field due to current Section 20.1

Example: Two Parallel Wires I I 1 2 What is the force on wire 1 due to wire 2? Is the force attractive or repulsive? r L

A current is a collection of moving charges The direction of the force is given by the right-hand rule 2 Magnetic Force on a Current Section 20.4

Ampère’s Law Ampère’s Law can be used to calculate the magnetic field when there is symmetry Similar to Gauss’ Law for electric fields Relates the magnetic field along a path to the electric current enclosed by the path Section 20.7

Ampère’s Law, cont. The magnetic field along a closed path is related to the current enclosed by that path μ o is the permeability of free space μ o = 4 π x T. m / A If B varies in magnitude or direction along the path, Ampère’s Law is not useful Section 20.7

Magnetic Field of a Long Straight Wire Ampère’s Law can be used to find the magnetic field near a long, straight wire Choose a circular, closed path B || is the same all along the path If the circular path has a radius r, then the total path length is 2 π r Applying Ampère’s Law gives Section 20.7

Example: Two Parallel Wires I I 1 2 What is the force on wire 1 due to wire 2? Is the force attractive or repulsive? r L

Field from a Current Loop It is not possible to find a simple path along which the magnetic field is constant Ampère’s Law cannot be easily applied From other techniques, Section 20.7

Field Inside a Solenoid By stacking many loops close together, the field along the axis is much larger than for a single loop A helical winding of wire is called a solenoid More practical than stacking single loops For a long solenoid, there is practically no field outside Section 20.7

Example: Long Solenoid What is the magnetic field, B, inside a “long” solenoid?

Torque on a Current Loop A magnetic field can produce a torque on a current loop The directions of the forces can be found from right-hand rule 2 Section 20.5

Torque, cont. On two sides, the current is parallel or antiparallel to the field, so the force is zero on those sides The forces on sides 1 and 3 are in opposite directions and produce a torque on the loop When the angle between the loop and the field is θ, the torque is τ = I L 2 B sin θ For different shapes, this becomes τ = I A B sin θ Section 20.5

Magnetic Moment, μ For a current loop, the magnetic moment, μ, is I A The direction of the magnetic moment is either along the axis of the bar magnet or perpendicular to the current loop The strength of the torque depends on the magnitude of the magnetic moment τ = μ B sin θ Section 20.5

Electric Motor A magnetic field can produce a torque on a current loop In a practical motor, a solenoid is used instead of a single loop Additional set-up is needed to keep the shaft rotating Electric generators are motors in reverse A generator produces an electric current by rotating a coil between the poles of the magnet Section 20.10

Magnetism Summary Magnetic Fields: (T) North and South poles Ampere’s Law Field from straight wire Right-hand rule 1 Field from current loop Field from Solenoids Magnetic Forces: (N) Force on moving charge Right-hand rule 2 Mass spectrometer Hall Effect Force on current wire Torque on current loop Magnetic moment

Chapter 21 Magnetic Induction

Electromagnetism Electric and magnetic phenomena were connected by Ørsted in 1820 He discovered an electric current in a wire can exert a force on a compass needle He concluded that a changing electric field will produce a magnetic field Can a magnetic field produce an electric field? Experiments were done by Michael Faraday Section 21.1

Faraday’s Experiment If the bar magnet was in motion, a current was observed If the magnet is stationary, the current and the electric field are both zero Same results occurred by moving the loop instead of the magnet Section 21.1

Faraday’s Experiment, cont. A solenoid is positioned inside a loop of wire When the current through the solenoid is constant, there is no current in the wire When the switch is opened or closed, current flows in the wire Section 21.1

Conclusions from Experiments Faraday’s experiments show that an electric current is produced in the wire loop only when the magnetic field through the loop is changing A changing magnetic field produces an electric field An electric field produced in this way is called an induced electric field The phenomena is called electromagnetic induction Section 21.1

Magnetic Induction Moving electric charges created a magnetic field Moving charges change the electric field A changing magnetic field created an electric field This effect is called magnetic induction This links electricity and magnetism in a fundamental way Magnetic induction is also the key to many practical applications WiTricity:

Magnetic Flux Faraday developed a quantitative theory of induction now called Faraday’s Law The law shows how to calculate the induced electric field in different situations Faraday’s Law uses the concept of magnetic flux Magnetic flux is similar to the concept of electric flux The magnetic flux is Section 21.2

Magnetic Flux, cont. If the field is perpendicular to the surface, Φ B = B A If the field makes an angle θ with the normal to the surface, Φ B = B A cos θ If the field is parallel to the surface, Φ B = 0 Section 21.2

Magnetic Flux, final The magnetic flux can be defined for any surface A complicated surface can be broken into small regions and the definition of flux applied The total flux is the sum of the fluxes through all the individual pieces of the surface The unit of magnetic flux is the Weber (Wb) 1 Wb = 1 T. m 2 Section 21.2

Faraday’s Law Faraday’s Law indicates how to calculate the potential difference that produces the induced current Written in terms of the emf (voltage) induced in the wire loop The induced voltage is proportional to the rate of change of the magnetic flux The negative sign is Lenz’s Law Section 21.2

Applying Faraday’s Law, cont. The emf is produced by changes in the magnetic flux through the circuit A constant flux does not produce an induced voltage The flux can change due to Changes in the magnetic field Changes in the area Changes in the angle The voltmeter will indicate the direction of the induced emf and induced current and electric field It is related to the electric field directly along and inside the wire loop Section 21.2

Faraday’s Law, Summary Only changes in the magnetic flux matter Rapid changes in the flux produce larger values of emf than do slow changes This dependency on frequency means the induced emf plays an important role in AC circuits The magnitude of the emf is proportional to the rate of change of the flux If the rate is constant, then the emf is constant In most cases, this isn’t possible and AC currents result Section 21.2

Flux Though a Changing Area What is the magnitude of the voltage induced in the wire loop? What is the power dissipated through the resistor? Section 21.2

Conservation of Energy The mechanical power put into the bar equals the electrical power delivered to the resistor Energy is converted from mechanical to electrical, but the total energy remains the same Conservation of energy is obeyed by electromagnetic phenomena Section 21.2

Changing a Magnetic Flux, Summary A change in magnetic flux can be produced in four ways If the magnitude of the magnetic field changes with time If the area changes with time If the loop rotates so that the angle changes with time If the loop moves from one region of magnetic field to another region of differing magnetic field Section 21.2

Lenz’s Law Lenz’s Law gives an easy way to determine the sign of the induced emf Determine direction of induced current Lenz’s Law states “the magnetic field produced by an induced current always opposes any changes in the magnetic flux” Section 21.3

Lenz’s Law, Example 1 Consider a metal loop in which the magnetic field passes upward through the loop and increases with time The magnetic field produced by the induced emf must oppose the change in flux Therefore, the induced magnetic field must be downward and the induced current will be clockwise Section 21.3

Lenz’s Law, Example 2 Consider a metal loop in which the magnetic field passes upward through the loop and decreases with time The magnetic field produced by the induced emf must oppose the change in flux Therefore, the induced magnetic field must be downward and the induced current will be counterclockwise Section 21.3

Example: Falling Magnet Suppose a bar magnet falls downward through a loop of wire, north pole first. What is the direction of the induced current? Suppose a bar magnet falls upward through a loop of wire, south pole first. What is the direction of the induced current? N v S v

Lenz’s Law and Conservation of Energy Mathematically, Lenz’s Law is just the negative sign in Faraday’s Law It is actually a consequence of conservation of energy Therefore, conservation of energy is contained in Faraday’s Law Physicists believe all laws of physics must (eventually) satisfy the principle of conservation of energy Section 21.3

Electrical Generator Uses changing flux to induce current in wire loop The wire loop has a fixed area, but is mounted on a rotating shaft The angle between the field and the plane of the loop changes as the loop rotates If the shaft rotates with a constant angular velocity, the flux varies sinusoidally with time Sinusoidal voltage is source of AC circuits Section 21.2