week 8 The Magnetic Field

General Principles

Applications

Start with magnetic forces on moving charges and currents

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) downwards 4) to the right 5) to the left x x x x x x v x x x x x x q x x x x x x

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) downwards 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 x x x x x x q x x F x x x x

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) downwards 4) upwards 5) to the left x x x x x x x x x x x x q x x x x x v x

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) downwards 4) upwards 5) to the left Using the right-hand rule, you can see that the magnetic force is directed upwards. Remember that the magnetic force must be perpendicular to BOTH the B field and the velocity. x x x x x x F x x x x x x q x x x x x v x

A positive charge enters a uniform magnetic field as shown. What is the direction of the magnetic force? (here R stands for pointing to the Right) 1) out of the page 2) into the page 3) zero 4) to the right 5) to the left v q

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 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. v F q

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

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 The charge is moving parallel to the magnetic field, so it does not experience any magnetic force. Remember that the magnetic force is given by: F = v B sin(θ). v q F = 0

An electron moves in the magnetic field of magnitude 0.50 T with a speed of 1.0 x 10 7 m/s in the direction shown in the figure. What is the magnetic force on the electron? Ans: 8.0 x 10-13 N, in direction of negative z-axis

A beam of atoms enters a magnetic field region. What path will the atoms follow? 1 4 2 3

A beam of atoms enters a magnetic field region. What path will the atoms follow? 1 4 2 3 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?

A proton beam enters into a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) y 3) + x 4) + z (out of page) 5) z (into page) y x

A proton beam enters into a magnetic field region as shown below. What is the direction of the magnetic field B? 1) + y 2) y 3) + x 4) + z (out of page) 5) z (into page) 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. y x Follow-up: What would happen to a beam of atoms?

The Hall voltage across a 1.0-mm-thick conductor in a 1.0 T magnetic field is 3.2 μv when the current is 15 A. What is the charge-carrier density in this conductor? Ans: 2.9 x 10 28 m -3 (Derive V=IB/e n t first)

Two particles of the same mass enter a magnetic field with the same speed and follow the paths shown. Which particle has the bigger charge? 3) both charges are equal 4) impossible to tell from the picture 1 2 The relevant equation for us is: According to this equation, the bigger the charge, the smaller the radius. Follow-up: What is the sign of the charges in the picture?

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.

What direction would a B field have to point for a beam of electrons moving to the right to go undeflected through a region where there is a uniform electric field pointing vertically upward? 1) up (parallel to E ) 2) down (antiparallel to E ) 3) into the page 4) out of the page 5) impossible to accomplish Without a B field, the electrons feel an electric force downwards. In order to compensate, the magnetic force has to point upwards. Using the right-hand rule and the fact that the electrons are negatively charged leads to a B field pointing out of the page. electrons B =? v E

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 I B

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. I B

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 I B

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 I When the current is parallel to the magnetic field lines, the force on the wire is zero. B

A rectangular current loop is in a uniform magnetic field. What is the direction of the net force on the loop? 1) + x 2) + y 3) zero 4) - x 5) - y z B x y

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

If there is a current in the loop in the direction shown, the loop will: 1) move up 2) move down 3) rotate clockwise 4) rotate counterclockwise 5) both rotate and move B field out of North B field into South N S S N

If there is a current in the loop in the direction shown, the loop will: 1) move up 2) move down 3) rotate clockwise 4) rotate counterclockwise 5) both rotate and move 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

A circular loop of wire carries a constant current. If the loop is placed in a region of uniform magnetic field, the net magnetic force on the loop is A. perpendicular to the plane of the loop, in a direction given by a right-hand rule B. perpendicular to the plane of the loop, in a direction given by a left-hand rule C. in the same plane as the loop D. zero E. answer depends on the magnitude and direction of the current and on the magnitude and direction of the magnetic field

A circular loop of wire carries a constant current. If the loop is placed in a region of uniform magnetic field, the net magnetic torque on the loop A. tends to orient the loop so that its plane is perpendicular to the direction of the magnetic field B. tends to orient the loop so that its plane is edge-on to the direction of the magnetic field C. tends to make the loop rotate around its axis D. is zero E. answer depends on the magnitude and direction of the current and on the magnitude and direction of the magnetic field

A square current loop 5.0 cm on each side carries a 500 ma current. The loop is in a 1.2 T uniform magnetic fields. The axis of the loop, perpendicular to the plane of the loop, is 30 o away from the field direction. What is the magnitude of the torque on the current loop? Ans: 7.5 10-4 N.m

Magnetic field due to a moving charge or current

General Principles

Applications

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 4 P 1 3 2

If the currents in these wires have the same magnitude, but opposite directions, what is the direction of the magnetic field at point 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. 1) direction 1 2) direction 2 3) direction 3 4) direction 4 5) the B field is zero 1 P 4 2 3

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 B=? 2 B=? 3 B=?

What are the magnetic field strength and direction at the dot in the figure? Ans: 2.83 x 10-16 T

Two long, straight wires are oriented perpendicular to the xy plane. They carry currents of magnitude I in opposite directions a shown. At point P, the magnetic field due to these currents A. is in the positive x direction B. is in the negative x direction C. is in the positive y direction D. is in the negative y direction E. none of the above

What are the magnetic field strength and direction at points a to c? Ans: Ba = Bc = 6.7 x 10-5 T out of page Bb = 2.0 x 10-4 T into page

Combine both

Applications

A positive charge moves parallel to a wire. If a current is suddenly turned on, which direction will the force act? 1) + z (out of page) 2) - z (into page) 3) + x 4) - x 5) - y y +q I z x

A positive charge moves parallel to a wire. If a current is suddenly turned on, which direction will the force act? 1) + z (out of page) 2) - z (into page) 3) + x 4) - x 5) - y 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. y +q x I z

Two positive point charges move side by side in the same direction with the same velocity. The magnetic force that the upper point charge exerts on the lower one A. is directed toward the upper point charge (that is, the force is attractive) B. is directed away from the upper point charge (that is, the force is repulsive) C. is in the direction of the velocity D. is opposite to the direction of the velocity E. none of the above

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). Follow-up: What happens when one of the currents is turned off?

Two long parallel wires separated by a distance d carry currents I1 and I2 in the same direction. What is the magnetic force per unit length between the wires? Ans: μ0 I1 I2 / 2 π d Note: This is used to define the Ampère and then the Coulomb.

What is the direction of the magnetic field at the center (point P) of the square loop of current? 1) left 2) right 3) zero 4) into the page 5) out of the page I P

What is the direction of the magnetic field at the center (point P) of the square loop of current? 1) left 2) right 3) zero 4) into the page 5) out of the page 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. I P

The figure shows, in cross section, three conductors that carry currents perpendicular to the plane of the figure. If the currents I 1, I 2, and I 3 all have the same magnitude, for which path(s) is the line integral of the magnetic field equal to zero? A. path a only B. paths a and c C. paths b and d D. paths a, b, c, and d E. answer depends on whether the integral goes clockwise or counterclockwise around the path

An application of Ampere s law: the toroid (you MUST have studied the solenoid) See problem 54 (ch 33) B = µ 0NI 2πr