Charges moving in a wire Up to this point we have focused our attention on PARTICLES or CHARGES only. The charges could be moving together in a wire. Thus, if the wire had a CURRENT (moving charges), it too will experience a force when placed in a magnetic field. You simply used the RIGHT HAND ONLY and the thumb will represent the direction of the CURRENT instead of the velocity.

Magnetic Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field The current is a collection of many charged particles in motion The direction of the force is given by right hand rule #1

Force on a Wire The blue x’s indicate the magnetic field is directed into the page The x represents the tail of the arrow Blue dots would be used to represent the field directed out of the page The • represents the head of the arrow In this case, there is no current, so there is no force

Force on a Wire, cont B is into the page The current is up the page Point your fingers into the page The current is up the page Point your thumb up the page The force is to the left Your palm should be pointing to the left

Force on a Wire, final B is into the page The current is down the page Point your fingers into the page The current is down the page Point your thumb down the page The force is to the right Your palm should be pointing to the right

Charges moving in a wire At this point it is VERY important that you understand that the MAGNETIC FIELD is being produced by some EXTERNAL AGENT

Force on a Wire, equation The magnetic force is exerted on each moving charge in the wire The total force is the sum of all the magnetic forces on all the individual charges producing the current F = B I ℓ sin θ θ is the angle between B and I The direction is found by the right hand rule, pointing your thumb in the direction of I instead of v

Example A 36-m length wire carries a current of 22A running from right to left. Calculate the magnitude and direction of the magnetic force acting on the wire if it is placed in a magnetic field with a magnitude of 0.50 x10-4 T and directed up the page. +y B = +y I = -x F = +z +x 0.0396 N -z, into the page

WHY does the wire move? The real question is WHY does the wire move? It is easy to say the EXTERNAL field moved it. But how can an external magnetic field FORCE the wire to move in a certain direction? THE WIRE ITSELF MUST BE MAGNETIC!!! In other words the wire has its own INTERNAL MAGNETIC FIELD that is attracted or repulsed by the EXTERNAL FIELD. As it turns out, the wire’s OWN internal magnetic field makes concentric circles round the wire.

A current carrying wire’s INTERNAL magnetic field To figure out the DIRECTION of this INTERNAL field you use the right hand rule. You point your thumb in the direction of the current then CURL your fingers. Your fingers will point in the direction of the magnetic field

The MAGNITUDE of the internal field The magnetic field, B, is directly proportional to the current, I, and inversely proportional to the circumference.

Example A long, straight wires carries a current of 5.00 A. At one instant, a proton, 4 mm from the wire travels at 1500 m/s parallel to the wire and in the same direction as the current. Find the magnitude and direction of the magnetic force acting on the proton due to the field caused by the current carrying wire. v X X X 2.51 x 10- 4 T 4mm + B = +z v = +y F = 6.02 x 10- 20 N -x 5A

Magnetic Fields – Long Straight Wire A current-carrying wire produces a magnetic field The compass needle deflects in directions tangent to the circle The compass needle points in the direction of the magnetic field produced by the current

Direction of the Field of a Long Straight Wire Right Hand Rule #2 Grasp the wire in your right hand Point your thumb in the direction of the current Your fingers will curl in the direction of the field

Magnitude of the Field of a Long Straight Wire The magnitude of the field at a distance r from a wire carrying a current of I is µo = 4  x 10-7 T m / A µo is called the permeability of free space

Ampère’s Law André-Marie Ampère found a procedure for deriving the relationship between the current in a arbitrarily shaped wire and the magnetic field produced by the wire Ampère’s Circuital Law B|| Δℓ = µo I Sum over the closed path

Ampère’s Law, cont Choose an arbitrary closed path around the current Sum all the products of B|| Δℓ around the closed path

Ampère’s Law to Find B for a Long Straight Wire Use a closed circular path The circumference of the circle is 2  r This is identical to the result previously obtained

Magnetic Force Between Two Parallel Conductors The force on wire 1 is due to the current in wire 1 and the magnetic field produced by wire 2 The force per unit length is:

Force Between Two Conductors, cont Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in the opposite directions repel each other

Defining Ampere and Coulomb The force between parallel conductors can be used to define the Ampere (A) If two long, parallel wires 1 m apart carry the same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere If a conductor carries a steady current of 1 A, then the quantity of charge that flows through any cross section in 1 second is 1 C

QUICK QUIZ 19.5 If I1 = 2 A and I2 = 6 A in the figure below, which of the following is true: (a) F1 = 3F2, (b) F1 = F2, or (c) F1 = F2/3?

QUICK QUIZ 19.5 ANSWER (b). The two forces are an action-reaction pair. They act on different wires, and have equal magnitudes but opposite directions.

Magnetic Force on Current-Carrying Wire F = I L B sin I: current in Amps L: length in meters B: magnetic field in Tesla : angle between current and field

Sample Problem What is the force on a 100 m long wire bearing a 30 A current flowing north if the wire is in a downward-directed magnetic field of 400 mT?

Sample Problem What is the magnetic field strength if the current in the wire is 15 A and the force is downward and has a magnitude of 40 N/m? What is the direction of the current?

Magnetic Fields… Affect moving charge Caused by moving charge! F = qvBsinq F = ILBsinq Hand rule is used to determine direction of this force. Caused by moving charge!

Magnetic Field for Long Straight Wire B = oI / (2r) o: 4  10-7 T m / A μo magnetic permeability of free space I: current (A) r: radial distance from center of wire (m)

Right Hand Rule for straight currents •  Curve your fingers Place your thumb (which is presumably pretty straight) in direction of current. Curved fingers represent curve of magnetic field. Field vector at any point is tangent to field line.

I For straight currents

Sample Problem What is the magnitude and direction of the magnetic field at point P, which is 3.0 m away from a wire bearing a 13.0 Amp current? P 3.0 m I = 13.0 A

Sample Problem – not in packet What is the magnitude and direction of the force exerted on a 100 m long wire that passes through point P which bears a current of 50 amps in the same direction? I2 = 50.0 A P 3.0 m I1 = 13.0 A

Principle of Superposition When there are two or more currents forming a magnetic field, calculate B due to each current separately and then add them together using vector addition.

Sample Problem 4.0 m 3.0 m I = 10.0 A P I = 13.0 A What is the magnitude and direction of the magnetic field at point P? I = 10.0 A 4.0 m P 3.0 m I = 13.0 A

Sample Problem 7.0 m I = 10.0 A I = 13.0 A Where is the magnetic field zero? I = 10.0 A 7.0 m I = 13.0 A

In the 4th Grade You learned that coils with current in them make magnetic fields. The iron nail was not necessary to cause the field; it merely intensified it. N S B I

Solenoid A solenoid is a coil of wire. When current runs through the wire, it causes the coil to become an “electromagnet”. Air-core solenoids have nothing inside of them. Iron-core solenoids are filled with iron to intensify the magnetic field.

Magnetic Field Inside a Solenoid B = on I o: 4  10-7 T m / A n: number of coils per unit length I: current (A)

Magnetic Field around Curved Current B

Right Hand Rule for magnetic fields around curved wires Curve your fingers. Place them along wire loop so that your fingers point in direction of current. Your thumb gives the direction of the magnetic field in the center of the loop, where it is straight. Field lines curve around and make complete loops. B I

Sample Problem An air-core 10 cm long is wrapped with copper wire that is 0.1 mm in diameter. What must the current be through the wire if a magnetic field of 20 mT is to be produced inside the solenoid?

Sample Problem What is the direction of the magnetic field produced by the current I at A? At B? I A B

Magnetic Field around Curved Current B

Sample Problem What is the magnetic field inside the air-core solenoid shown if the resistance of the copper wire is assumed to be negligible? There are 100 windings per cm. Identify the north pole. 120 V I 100-W

Magnetic Field of a Current Loop The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop All the segments, Δx, contribute to the field, increasing its strength

Magnetic Field of a Current Loop – Total Field

Magnetic Field of a Solenoid If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid It is also known as an electromagnet since it acts like a magnet only when it carries a current

Magnetic Field of a Solenoid, 2 The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together This indicates that the field inside the solenoid is nearly uniform and strong The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

Magnetic Field in a Solenoid, 3 The field lines of the solenoid resemble those of a bar magnet

Magnetic Field in a Solenoid, Magnitude The magnitude of the field inside a solenoid is constant at all points far from its ends B = µo n I n is the number of turns per unit length n = N / ℓ The same result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field in a Solenoid from Ampère’s Law A cross-sectional view of a tightly wound solenoid If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero Apply Ampère’s Law to the red dashed rectangle

Magnetic Effects of Electrons -- Orbits An individual atom should act like a magnet because of the motion of the electrons about the nucleus Each electron circles the atom once in about every 10-16 seconds This would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom

Magnetic Effects of Electrons – Orbits, cont The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

Magnetic Effects of Electrons -- Spins Electrons also have spin The classical model is to consider the electrons to spin like tops It is actually a relativistic and quantum effect

Magnetic Effects of Electrons – Spins, cont The field due to the spinning is generally stronger than the field due to the orbital motion Electrons usually pair up with their spins opposite each other, so their fields cancel each other That is why most materials are not naturally magnetic

Magnetic Effects of Electrons -- Domains In some materials, the spins do not naturally cancel Such materials are called ferromagnetic Large groups of atoms in which the spins are aligned are called domains When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others This causes the material to become magnetized

Domains, cont Random alignment, a, shows an unmagnetized material When an external field is applied, the domains aligned with B grow, b

Domains and Permanent Magnets In hard magnetic materials, the domains remain aligned after the external field is removed The result is a permanent magnet In soft magnetic materials, once the external field is removed, thermal agitation cause the materials to quickly return to an unmagnetized state With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field