Lecture 14 Magnetism

Magnets... two poles: N and S Like poles repel Unlike poles attract

Magnetic Field lines: (defined in same way as electric field lines, direction and density)

Broken Permanent Magnet If we break a permanent magnet in half, we do not get a separate north pole and south pole. When we break a bar magnet in half, we always get two new magnets, each with its own north and south pole.

Source of Magnetic Fields What is the source of magnetic fields? Answer: electric charge in motion e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of magnets. Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect)

Magnetic Materials Materials can be classified by how they respond to an applied magnetic field, B app. Paramagnetic (aluminum, tungsten, oxygen,…) Atomic magnetic dipoles (~atomic bar magnets) tend to line up with the field, increasing it. But thermal motion randomizes their directions, so only a small effect persists: B ind ~ B app Diamagnetic (gold, copper, water,…) The applied field induces an opposing field; again, this is usually very weak; B ind ~ -B app [Exception: Superconductors exhibit perfect diamagnetism  they exclude all magnetic fields] Ferromagnetic (iron, cobalt, nickel,…) Somewhat like paramagnetic, the dipoles prefer to line up with the applied field. But there is a complicated collective effect due to strong interactions between neighboring dipoles  they tend to all line up the same way. Very strong enhancement. B ind ~ B app 10 +5

Magnetic Field Direction A vector quantity: magnitude and direction… The letter B is used to represent magnetic fields.

Magnetic Field of the Earth A small magnetic bar should be said to have north and south seeking poles. The north of the bar points towards the North of the Earth. The geographic north corresponds to a south magnetic pole and the geographic south corresponds to a magnetic north. The configuration of the Earth magnetic resemble that of a (big) magnetic bar one would put in its center.

Magnetic Field of the Earth

Magnetic Fields in analogy with Electric Fields Electric Field: Distribution of charge creates an electric field E in the surrounding space. Distribution of charge creates an electric field E in the surrounding space. Field exerts a force F=q E on a charge q Field exerts a force F=q E on a charge q Magnetic Field: Moving charge or current creates a magnetic field B in the surrounding space. Moving charge or current creates a magnetic field B in the surrounding space. Field exerts a force F on a charge moving q Field exerts a force F on a charge moving q

The magnetic force Observations show that the force is proportional to The field The field The charge The charge The velocity of the particle The velocity of the particle The sine of the angle between the field and the direction of the particle’s motion. The sine of the angle between the field and the direction of the particle’s motion.

Strength and direction of the Magnetic Force on a charge in motion +q F B v [Q] The figure shows a charged particle with velocity v travels through a uniform magnetic field B. What is the direction of the magnetic force F on the particle?

Magnetic Field Magnitude [Q] An electron (q = -1.6x C) is moving at 3 x 10 5 m/s in the positive x direction. A magnetic field of 0.8T is in the positive z direction. The magnetic force on the electron is:

Magnetic Field Units [F] = Newton [v] = m/s [q] = C [B] = tesla (T). Also called weber (Wb) per square meter. Also called weber (Wb) per square meter. 1 T = 1 Wb/m 2. 1 T = 1 Wb/m 2. 1 T = 1 N s m -1 C T = 1 N s m -1 C T = 1 N A -1 m T = 1 N A -1 m -1. CGS unit is the Gauss (G) 1 T = 10 4 G. 1 T = 10 4 G.

Right Hand Rule

Example: Proton traveling in Earth’s magnetic field. A proton moves with a speed of 1.0 x 10 5 m/s through the Earth’s magnetic field which has a value of 55  T a particular location. When the proton moves eastward, the magnetic force acting on it is a maximum, and when it moves northward, no magnetic force acts on it. What is the strength of the magnetic force? And what is the direction of the magnetic field? Northward or southward.

Magnetic Force on Current- carrying conductor.  A magnetic force is exerted on a single charge in motion through a magnetic field.  That implies a force should also be exerted on a collection of charges in motion through a conductor I.e. a current.  The force on a current is the sum of all elementary forces exerted on all charge carriers in motion.

Magnetic Force on Current If B is directed into the page we use blue crosses representing the tail of arrows indicating the direction of the field, If B is directed out of the page, we use dots. If B is in the page, we use lines with arrow heads. x x x x x x x x x x x x x x x x x

Force on a wire carrying current in a magnetic field. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x I = 0 I B in x x x x x x x x x x x x x x x x x I B in

x x x x x x x x x x x x x x x x x x x x x Force on a wire carrying current in a magnetic field. A vdvd q i vdvd F L B (Out of the paper)...

Force on a wire carrying current in a magnetic field. General Case: field at angle  relative to current. I B  B sin 

Example : Wire in Earth’s B Field A wire carries a current of 22 A from east to west. Assume that at this location the magnetic field of the earth is horizontal and directed from south to north, and has a magnitude of 0.50 x T. Find the magnetic force on a 36-m length of wire. What happens if the direction of the current is reversed?

Motion of Charged Particle in magnetic field Consider positively charge particle moving in a uniform magnetic field. Suppose the initial velocity of the particle is perpendicular to the direction of the field. Then a magnetic force will be exerted on the particle and make follow a circular path.    F v q r B in

The magnetic force produces a centripetal acceleration. The particle travels on a circular trajectory with a radius: What is the period of revolution of the motion? The frequency

Example : Proton moving in uniform magnetic field A proton is moving in a circular orbit of radius 14 cm in a uniform magnetic field of magnitude 0.35 T, directed perpendicular to the velocity of the proton. Find the orbital speed of the proton. Use

Example: If a proton moves in a circle of radius 21 cm perpendicular to a B field of 0.4 T, what is the speed of the proton and the frequency of motion? v r xx xx

[Q]: Consider the mass spectrometer. The electric field between the plates of the velocity selector is 950 V/m, and the magnetic fields in both the velocity selector and the deflection chamber have magnitudes of T. Calculate the radius of the path in the system for a singly charged ion with mass m=2.18× kg.