 # Magnetic Fields and Forces

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Magnetic Fields and Forces

The Definition of B: We can define a magnetic field, B, by firing a charged particle through the point at which is to be defined, using various directions and speeds for the particle and determining the force that acts on the particle at that point. B is then defined to be a vector quantity that is directed along the zero-force axis. The magnetic force on the charged particle, FB, is defined to be: Here q is the charge of the particle, v is its velocity, and B the magnetic field in the region. The magnitude of this force is then: Here f is the angle between vectors v and B.

Finding the Magnetic Force on a Particle:

The Definition of B: The SI unit for B that follows is newton per coulomb-meter per second. For convenience, this is called the tesla (T): An earlier (non-SI) unit for B is the gauss (G), and

Current-Carrying Wire:
Magnetic Force on a Current-Carrying Wire: Here L is a length vector that has magnitude L and is directed along the wire segment in the direction of the (conventional) current. If a wire is not straight or the field is not uniform, we can imagine the wire broken up into small straight segments . The force on the wire as a whole is then the vector sum of all the forces on the segments that make it up. In the differential limit, we can write and we can find the resultant force on any given arrangement of currents by integrating over that arrangement.

Torque on a Current Loop:
To define the orientation of the loop in the magnetic field, we use a normal vector n that is perpendicular to the plane of the loop. For side 2 the magnitude of the force acting on this side is F2=ibB sin(90°-q)=ibB cosq =F F2 and F4 cancel out exactly. Forces F1 and F3 have the common magnitude iaB. As Fig c shows, these two forces do not share the same line of action; so they produce a net torque. For N loops, when A=ab, the area of the loop, the total torque is:

TOP VIEW F q n B x q Fsinq Fsinq F=IaB q

TOP VIEW n B F=IaB b F = 2*(b/2)Fsinq = IbaBsinq = IABsinq Fsinq Fsinq
x q Fsinq F=IaB Fsinq b q = 2*(b/2)Fsinq = IbaBsinq = IABsinq

Torque on a Current Loop
= IABsinq If multiple (N) Loops = NIABsinq If multiple Circular Loops = NIpr2Bsinq

The Magnetic Dipole Moment, m:
Definition: Here, N is the number of turns in the coil, i is the current through the coil, and A is the area enclosed by each turn of the coil. Direction: The direction of m is that of the normal vector to the plane of the coil. The definition of torque can be rewritten as: Just as in the electric case, the magnetic dipole in an external magnetic field has an energy that depends on the dipole’s orientation in the field: A magnetic dipole has its lowest energy (-mB cos 0=-mB) when its dipole moment m is lined up with the magnetic field. It has its highest energy (-mB cos 180°=+mB) when m is directed opposite the field.

ConcepTest 22.1a Magnetic Force I
A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force? 1) out of the page 2) into the page 3) downward 4) to the right 5) to the left Using the right-hand rule, you can see that the magnetic force is directed to the left. Remember that the magnetic force must be perpendicular to BOTH the B field and the velocity. x x x x x x v q x x x x x x v q F

ConcepTest 22.1b Magnetic Force II
A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force? 1) out of the page 2) into the page 3) downward 4) upward 5) to the left Using the right-hand rule, you can see that the magnetic force is directed upward. Remember that the magnetic force must be perpendicular to BOTH the B field and the velocity. x x x x x x v q x x x x x x v q F

ConcepTest 22.1c Magnetic Force III
A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force? 1) out of the page 2) into the page 3) zero 4) to the right 5) to the left ® ® ® ® ® v q ® ® ® ® ® v q F Using the right-hand rule, you can see that the magnetic force is directed into the page. Remember that the magnetic force must be perpendicular to BOTH the B field and the velocity.

ConcepTest 22.2 Atomic Beams
A beam of atoms enters a magnetic field region. What path will the atoms follow? x x x x x x x x x x x x 1 2 3 4 Atoms are neutral objects whose net charge is zero. Thus they do not experience a magnetic force. Follow-up: What charge would follow path #3? What about path #1?

ConcepTest 22.3 Magnetic Field
1) + y 2) – y 3) + x 4) + z (out of page) 5) – z (into page) A proton beam enters into a magnetic field region as shown below. What is the direction of the magnetic field B? The picture shows the force acting in the +y direction. Applying the right-hand rule leads to a B field that points into the page. The B field must be out of the plane because B  v and B  F. x y

ConcepTest 22.4b Mass Spectrometer
A proton enters a uniform magnetic field that is perpendicular to the proton’s velocity. What happens to the kinetic energy of the proton? 1) it increases 2) it decreases 3) it stays the same 4) depends on the velocity direction 5) depends on the B field direction The velocity of the proton changes direction but the magnitude (speed) doesn’t change. Thus the kinetic energy stays the same. x x x x x x x x x x x x

ConcepTest 22.6a Magnetic Force on a Wire I
A horizontal wire carries a current and is in a vertical magnetic field. What is the direction of the force on the wire? 1) left 2) right 3) zero 4) into the page 5) out of the page Using the right-hand rule, we see that the magnetic force must point out of the page. Since F must be perpendicular to both I and B, you should realize that F cannot be in the plane of the page at all. B I

ConcepTest 22.6b Magnetic Force on a Wire II
A horizontal wire carries a current and is in a vertical magnetic field. What is the direction of the force on the wire? 1) left 2) right 3) zero 4) into the page 5) out of the page B I When the current is parallel to the magnetic field lines, the force on the wire is zero.

ConcepTest 22.7a Magnetic Force on a Loop I
1) + x 2) + y 3) zero 4) - x 5) - y A rectangular current loop is in a uniform magnetic field. What is the direction of the net force on the loop? B x z y Using the right-hand rule, we find that each of the four wire segments will experience a force outward from the center of the loop. Thus, the forces of the opposing segments cancel, so the net force is zero.

Magnetic Fields from Wires…
Biot-Savart Law (today) and Ampere’s Law

29.2: Calculating the Magnetic Field due to a Current
The magnitude of the field dB produced at point P at distance r by a current length element i ds turns out to be where q is the angle between the directions of and , a unit vector that points from ds toward P. Symbol m0 is a constant, called the permeability constant, whose value is Therefore, in vector form

29.2: Magnetic Field due to a Long Straight Wire:

ConcepTest 22.10 Current Loop
1) left 2) right 3) zero 4) into the page 5) out of the page What is the direction of the magnetic field at the center (point P) of the square loop of current? P I Use the right-hand rule for each wire segment to find that each segment has its B field pointing out of the page at point P.

ConcepTest 22.8a Magnetic Field of a Wire I
If the currents in these wires have the same magnitude but opposite directions, what is the direction of the magnetic field at point P? 1) direction 1 2) direction 2 3) direction 3 4) direction 4 5) the B field is zero 1 P Using the right-hand rule, we can sketch the B fields due to the two currents. Adding them up as vectors gives a total magnetic field pointing downward. 4 2 3

ConcepTest 22.8b Magnetic Field of a Wire II
Each of the wires in the figures below carry the same current, either into or out of the page. In which case is the magnetic field at the center of the square greatest? 1) arrangement 1 2) arrangement 2 3) arrangement 3 4) same for all 1 2 3

Some examples: Magnetic Field due to a Long Straight Wire

Another example: Magnetic Field due to a Current in a Circular Arc of Wire

The equation for the magnetic field of a straight, current
carrying wire is given by , but the magnetic field at the center of a single closed circular loop is given by Although these equations look similar, there is an important difference between these two equations, other that the factor of . What is it? a) The µ0 factor is different for the two situations. b) The variable R represents two different lengths. c) The i represents two different types of current.

ConcepTest 22.9a Field and Force I
A positive charge moves parallel to a wire. If a current is suddenly turned on, in which direction will the force act? 1) + z (out of page) 2) - z (into page) 3) + x 4) - x 5) - y z y x I +q Using the right-hand rule to determine the magnetic field produced by the wire, we find that at the position of the charge +q (to the left of the wire) the B field points out of the page. Applying the right-hand rule again for the magnetic force on the charge, we find that +q experiences a force in the +x direction.

ConcepTest 22.9b Field and Force II
Two straight wires run parallel to each other, each carrying a current in the direction shown below. The two wires experience a force in which direction? 1) toward each other 2) away from each other 3) there is no force The current in each wire produces a magnetic field that is felt by the current of the other wire. Using the right-hand rule, we find that each wire experiences a force toward the other wire (i.e., an attractive force) when the currents are parallel (as shown).

Force Between Two Parallel Wires:

Two parallel wires have currents that have the same direction, but differing magnitude. The current in wire A is i; and the current in wire B is 2i. Which one of the following statements concerning this situation is true? a) Wire A attracts wire B with half the force that wire B attracts wire A. b) Wire A attracts wire B with twice the force that wire B attracts wire A. c) Both wires attract each other with the same amount of force. d) Wire A repels wire B with half the force that wire B attracts wire A. e) Wire A repels wire B with twice the force that wire B attracts wire A.

ConcepTest 22.7b Magnetic Force on a Loop II
1) move up 2) move down 3) rotate clockwise 4) rotate counterclockwise 5) both rotate and move If there is a current in the loop in the direction shown, the loop will: Look at the North Pole: here the magnetic field points to the right and the current points out of the page. The right-hand rule says that the force must point up. At the south pole, the same logic leads to a downward force. Thus the loop rotates clockwise. F N S F

The drawing shows two long, straight wires that are parallel to each other and carry a current of magnitude i toward you. The wires are separated by a distance d; and the centers of the wires are a distance d from the y axis. Which one of the following expressions correctly gives the magnitude of the total magnetic field at the origin of the x, y coordinate system? a) b) c) d) e) zero tesla

Ampere’s Law

Ampere’s Law:

Ampere’s Law, Magnetic Field Outside a Long Straight Wire
Carrying Current:

A hollow cylindrical conductor (inner radius = a, outer radius = b) carries a current i uniformly spread over its cross section. Which graph below correctly gives B as a function of the distance r from the center of the cylinder?

Magnetic Field Inside a Long Straight Wire Carrying Current:

Solenoids and Toroids:

Solenoids: Here n is the number of turns per unit length of the solenoid

A copper cylinder has an outer radius 2R and an inner radius of R and carries a current i. Which one of the following statements concerning the magnetic field in the hollow region of the cylinder is true? a) The magnetic field within the hollow region may be represented as concentric circles with the direction of the field being the same as that outside the cylinder. b) The magnetic field within the hollow region may be represented as concentric circles with the direction of the field being the opposite as that outside the cylinder. c) The magnetic field within the hollow region is parallel to the axis of the cylinder and is directed in the same direction as the current. d) The magnetic field within the hollow region is parallel to the axis of the cylinder and is directed in the opposite direction as the current. e) The magnetic field within the hollow region is equal to zero tesla.

The drawing shows a rectangular wire loop that has one side passing through the center of a solenoid. Which one of the following statements describes the force, if any, that acts on the rectangular loop when a current is passing through the solenoid. a) The magnetic force causes the loop to move upward. b) The magnetic force causes the loop to move downward. c) The magnetic force causes the loop to move to the right. d) The magnetic force causes the loop to move to the left. e) The loop is not affected by the current passing through the solenoid or the magnetic field resulting from it.

A solenoid carries current I as shown in the figure
A solenoid carries current I as shown in the figure. If the observer could “see” the magnetic field inside the solenoid, how would it appear?

The diagrams show three circuits consisting of concentric circular arcs
(either half or quarter circles of radii r, 2r, and 3r) and radial lengths. The circuits carry the same current. Rank them according to the magnitudes of the magnetic fields they produce at C, least to greatest. 1, 2, 3 B) 3, 2, 1 C) 1, 3, 2 D) 2, 3, 1 E) 2, 1, 3

Application: The Hall Effect
When the sign of the charge which carries current changes, the magnetic force FB does not change. Thus, measuring the Hall voltage difference between the top (c) and bottom (a) of the conducting slab, reveals the sign of the charge carrier q and its density n. Conductor in B-field Demo: Hall probe VH