# Chapter 28. Magnetic Field

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Chapter 28. Magnetic Field
28.1. What is Physics?       28.2. What Produces a Magnetic Field?       28.3. The Definition of   Magnetic Field      28.4. Crossed Fields: Discovery of the Electron       28.5. Crossed Fields: The Hall Effect       28.6. A Circulating Charged Particle      28.7. Cyclotrons and Synchrotrons       28.8. Magnetic Force on a Current-Carrying Wire       28.9. Torque on a Current Loop       The Magnetic Dipole Moment

What is Physics? Permanent magnets
A magnet has two poles, a north pole and a south pole No Magnetic monopole available in nature.

The magnetic forces Like poles repel each other, and unlike poles attract.

The magnetic field A magnetic field exists in the region around a magnet. The magnetic field is a vector that has both magnitude and direction. The direction of the magnetic field at any point in space is the direction indicated by the north pole of a small compass needle placed at that point.

The magnetic field line
The lines originate from the north pole and end on the south pole; they do not start or stop in midspace. The magnetic field at any point is tangent to the magnetic field line at that point. The strength of the field is proportional to the number of lines per unit area that passes through a surface oriented perpendicular to the lines. The magnetic field lines will never come to cross each other.

What Produces a Magnetic Field?
Moving electrically charged particles, such as a current, produce a magnetic field Permanent magnet. Elementary particles such as electrons have an intrinsic magnetic field around them. The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. Such addition is the reason why a permanent magnet has a permanent magnetic field. In other materials, the magnetic fields of the electrons cancel out, giving no net magnetic field surrounding the material

Magnetic force on a Charged Particle
When a charge is placed in a magnetic field, it experiences a magnetic force if two conditions are met: The charge must be moving. No magnetic force acts on a stationary charge. The velocity of the moving charge must have a component that is perpendicular to the direction of the field.

Magnetic force on a Charged Particle
Right-Hand Rule The force       acting on a charged particle moving with velocity      through a magnetic field      is always perpendicular to      and     .

Two particles, having the same charge but different velocities, are moving in a constant magnetic field (see the drawing, where the velocity vectors are drawn to scale). Which particle, if either, experiences the greater magnetic force? (a) Particle 1 experiences the greater force, because it is moving perpendicular to the magnetic field. (b) Particle 2 experiences the greater force, because it has the greater speed. (c) Particle 2 experiences the greater force, because a component of its velocity is parallel to the magnetic field. (d) Both particles experience the same magnetic force, because the component of each velocity that is perpendicular to the magnetic field is the same.

Example 1 Magnetic Forces on Charged Particles
A proton in a particle accelerator has a speed of 5.0×106 m/s. The proton encounters a magnetic field whose magnitude is 0.40 T and whose direction makes an angle of θ =30.0° with respect to the proton’s velocity. Find (a) the magnitude and direction of the magnetic force on the proton and (b) the acceleration of the proton. The Mass of proton is 1.67x10-27 kg.

The Definition of Magnetic Field
The magnetic field B is a vector, and its direction is direction is along the zero-force axis. The magnitude B of the magnetic field at any point in space is defined as where F is the magnitude of the magnetic force on a positive test charge q0 , v is the velocity of the charge which makes an angle  with the direction of the magnetic field. SI Unit of Magnetic Field:

Differences of ELECTRIC AND MAGNETIC FIELDS
Direction of forces The electric force on a charged particle (both moving and stationary) is always parallel (or anti-parallel) to the electric field direction. The magnetic force on a moving charged particle is always perpendicular to both magnetic field and velocity of the particle. No magnetic force on a stationary charged particle.

THE WORK DONE ON A CHARGED PARTICLE:
The electric force can do work on the particle. The magnetic force cannot do work and change the kinetic energy of the charged particle.

Crossed Fields: Discovery of the Electron
1. Set and to zero and note the position of the spot on screen S due to the undeflected beam. 2. Turn on and measure the resulting beam deflection y. 3.  Maintaining , now turn on and adjust its value until the beam returns to the undeflected position. (With the forces in opposition, they can be made to cancel.)

Crossed Fields: The Hall Effect
Can the drifting conduction electrons in a copper wire also be deflected by a magnetic field? the magnitude of that potential difference is When the electric and magnetic forces are in balance

Sample Problem 28 Figure 28-9 shows a solid metal cube, of edge length d=1.5cm, moving in the positive y direction at a constant velocity of magnitude 4.0 m/s. The cube moves through a uniform magnetic field of magnitude T in the positive z direction. (a) Which cube face is at a lower electric potential and which is at a higher electric potential because of the motion through the field? (b) What is the potential difference between the faces of higher and lower electric potential?

The motion of a charged particle in a constant magnetic field
A charged particle in a constant magnetic field will do uniform circular motion The radius of the circle is

Three particles have identical charges and masses. They enter a constant magnetic field and follow the paths shown in the drawing. Rank the speeds of the particles, largest to smallest.

Example 3 The Motion of a Proton
A proton is released from rest at point A, which is located next to the positive plate of a parallel plate capacitor (see Figure 21.13). The proton then accelerates toward the negative plate, leaving the capacitor at point B through a small hole in the plate. The electric potential of the positive plate is 2100 V greater than that of the negative plate, so VA–VB=2100 V. Once outside the capacitor, the proton travels at a constant velocity until it enters a region of constant magnetic field of magnitude 0.10 T. The velocity is perpendicular to the magnetic field, which is directed out of the page in Figure Find (a) the speed vB of the proton when it leaves the negative plate of the capacitor, and (b) the radius r of the circular path on which the proton moves in the magnetic field.

Helical Paths If the velocity of a charged particle has a component parallel to the (uniform) magnetic field , the particle will move in a helical path about the direction of the field vector.

The Auroral

The Cyclotron

Proton Synchrotron

Magnetic Force on a Current-Carrying Wire

Magnetic Force on a Current-Carrying Wire
If the magnetic field is not perpendicular to the wire, as in Fig, the magnetic force is given by                                                                                                  Define     as a length vector that has magnitude L and is directed along the wire segment in the direction of the (conventional) current.

The same current-carrying wire is placed in the same magnetic field B in four different orientations (see the drawing). Rank the orientations according to the magnitude of the magnetic force exerted on the wire, largest to smallest.

Sample Problem 28 A straight, horizontal length of copper wire has a current i=28 A through it. What are the magnitude and direction of the minimum magnetic field needed to suspend the wire—that is, to balance the gravitational force on it? The linear density (mass per unit length) of the wire is 46.6 g/m.

Torque on a Current Loop
the vector   , which is normal to the plane of the coil with direction determined by right hand rule shown in fig. (b). A is area of loop

When a current-carrying loop is placed in a magnetic field, the loop tends to rotate such that its normal becomes aligned with the magnetic field

Torque on a Current coil
A coil containing N loops, each of area A, the force on each side is N times larger, and the torque on the coil becomes:

The Magnetic Dipole Moment of Coil
The direction of is that of the normal vector to the plane of the coil and thus is given by the same right-hand rule. That is, grasp the coil with the fingers of your right hand in the direction of current i; the outstretched thumb of that hand gives the direction of . The magnitude of is given by 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.

Magnetic Dipole in a Magnetic Field
The torque on the coil due to a magnetic field                                                                                The magnetic potential energy If an applied torque to rotates a magnetic dipole from an initial orientation   to another orientation   , then work Wa is done on the dipole by the applied torque is

Example The Torque Exerted on a Current-Carrying Coil
A coil of wire has an area of 2.0×10–4 m2, consists of 100 loops or turns, and contains a current of A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil.

A DC electric motor

Conceptual Questions A charged particle, passing through a certain region of space, has a velocity whose magnitude and direction remain constant. (a) If it is known that the external magnetic field is zero everywhere in this region, can you conclude that the external electric field is also zero? Explain. (b) If it is known that the external electric field is zero everywhere, can you conclude that the external magnetic field is also zero? Explain. Suppose that the positive charge in Figure 21.10a were launched from the negative plate toward the positive plate, in a direction opposite to the electric field. A sufficiently strong electric field would prevent the charge from striking the positive plate. Suppose the positive charge in Figure 21.10b were launched from the south pole toward the north pole, in a direction opposite to the magnetic field. Would a sufficiently strong magnetic field prevent the charge from reaching the north pole? Account for your answer.

Three particles move through a constant magnetic field and follow the paths shown in the drawing. Determine whether each particle is positively charged, negatively charged, or neutral. Give a reason for each answer. When one end of a bar magnet is placed near a TV screen, the picture becomes distorted. Why?

A positive charge moves along a circular path under the influence of a magnetic field. The magnetic field is perpendicular to the plane of the circle, as in Figure If the velocity of the particle is reversed at some point along the path, will the particle retrace its path? If not, draw the new path. Explain.