Magnetic Fields and Forces Chapter 20 Magnetic Fields and Forces

Review Magnetic fields Are due to moving charges Right-hand rule 1 Act similar to electric fields with important differences Field lines make loops from north to south poles No magnetic mono-poles Force is perpendicular to direction of field Exert a force on moving charges Right-hand rule 2

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

Right-hand Rule 1 What is the direction of the magnetic field in regions 1, 2 and 3? B1 and B3 are non-zero They would be zero if these were two sheets of current 1 I 2 I 3

Right-hand Rule 1

Right-hand Rule 2 What is the direction of the force on a charge moving perpendicular to an external magnetic field? What is the radius of the circle formed by the charge’s trajectory? Section 20.3

Mass Spectrometer Uses magnetic forces to separate particles by mass and charge Ions with different masses will travel in arcs with different radii Used to find the composition of a material Calculate charge to mass ratio Section 20.6

Example: Bubble Chamber Used in particle physics to investigate decay products What is the sign of the charge? What is the relative mass?

Hall Effect (+) charge moving to the right and (-) charge moving to the left produce the same current direction The Hall Effect can distinguish between the two options Section 20.6

Hall Effect, cont. For a current carrying wire in a perpendicular magnetic field, net charge will accumulate on the top and bottom of the wire Measuring the potential difference between top and bottom will indicate the sign of the charge carriers Section 20.6

Example: Hall Effect I V d What is the magnitude and direction of the magnetic field? ncopper = 8.47 x 1028 m-3 Side View I V Front View d

Magnetic Force on a Current An electric current is a collection of moving charges Obeys magnetic force law From the equation of the force on a moving charge, the force on a current-carrying wire is The direction of the force is given by the right-hand rule 2 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 μo is the permeability of free space μo = 4 π x 10-7 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 What is the force on wire 1 due to wire 2? Is the force attractive or repulsive? (Demonstration) I 1 L r 2 I

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 Assume a square loop with sides of length L carrying a current I in a constant magnetic field 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 L2 B sin θ For different shapes, this becomes τ = I A B sin θ Section 20.5

Magnetic Moment, μ τ = μ B sin θ 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 If the loop is attached to a rotating shaft, an electric motor is formed In a practical motor, a solenoid is used instead of a single loop Additional set-up is needed to keep the shaft rotating Section 20.10

Electric Generator Electric generators are closely related to motors A generator produces an electric current by rotating a coil between the poles of the magnet A motor in reverse Section 20.10