- Sketch the magnetic field lines of a bar magnet. Label the north and south poles and the direction of the magnetic field inside and outside of the magnet.
- Sketch the magnetic field lines of a long, straight current-carrying wire for two cases: when the wire is vertical and when the wire is coming in or out of the page. Label the direction of current and the direction of a magnetic field.
- Sketch magnetic field lines of a wire loop for two cases: when the loop is in the plane of the page and when the loop is horizontal. Label the direction of the current in the loop and the direction of the magnetic field.
- Sketch the magnetic field lines of a solenoid. Label the direction of the current in the solenoid and the direction of the magnetic field lines inside and outside of the solenoid. Compare this magnetic field to the bar magnet. How would the field look like for an ideal solenoid?

The magnetic field inside of a bar magnet is directed from its south pole to its north pole. As the field lines leave the magnet on the north side, they loop around outside and enter the magnet on the south side.

The magnetic field lines of a long, straight, current-carrying wire form concentric circles around the wire in a plane that is perpendicular to the wire. The direction of the field is determined by the right-hand rule: if you curve the fingers of your right hand in the direction of the field, your thumb will point in the direction of the current, and vice versa. The magnitude of the field is inversely proportional to the distance from the wire: . Here, is the current, is the distance to the wire, and is the magnetic permeability of vacuum.

Magnetic field lines of a wire loop are perpendicular to the plane of the loop inside the loop. They are nearly straight lines near the center and have greater curvature near the circumference of the loop. The direction of the field is determined by the right-hand rule: if you curve the fingers of your right hand in the direction of the current in the loop, your thumb will point in the direction of the field near the center, and vice versa. The magnetic field at the center of the loop is , where is the radius of the loop.

The magnetic field of a solenoid is the vector sum of the fields of several coaxial wire loops. The field lines are nearly straight inside the solenoid and loop around far away from the solenoid. They are similar to the field lines of a permanent magnet. For an ideal, infinitely-long solenoid, the magnetic field is zero on the outside and uniform inside: . Here, is the number density of the loops, and the number of loops per unit length of the solenoid is .

Review magnetic fields in section 22.3 of *College Physics* and Magnetic Field Created by Current in a Wire.

- What quantities does the force on a particle moving in a magnetic field depend on? How can you determine the direction of the force?
- What quantities does the magnetic force on a current-carrying wire depend on? How can you determine the direction of the force?
- Describe the interaction between the two long current-carrying wires. When do they repel, and when do they attract?
- What is the magnitude of the torque exerted on a current-carrying loop placed in a magnetic field? What orientation of the loop relative to the field results in maximum torque?

The magnetic force on a moving charged particle, sometimes also called the Lorentz force, is determined by the charge and velocity of the particle, and the magnetic field: . The force equals the charge of the particle times the vector product of the velocity of the particle and the magnetic field. From the definition of the vector product, this means that the magnitude of the force is , where is the angle between the velocity and the magnetic field. The direction of the force, for the positively-charged particle, is determined by the right-hand rule and is perpendicular to both vectors. For the negatively-charged particle, the force will be in the opposite direction.

To review the properties of vector products of two vectors, watch Cross Product Part I and Part II. The force on a charge moving in a magnetic field is discussed in Magnetic Force on a Charge and in section 22.4 of *College Physics*.

The magnetic force on a current-carrying wire can be found as the sum of the forces on all moving charges creating the current in the wire. For a straight wire of length , the magnitude of this force is , where is the current. The direction of the force is determined using the right-hand rule: as you curve the fingers of your right hand from the direction of the current toward the magnetic field vector, your thumb will point in the direction of the force. Read The Lorentz force and section 22.7 of *College Physics*, and watch Magnetic Force on a Current-Carrying Wire.

In the particular case of a long straight wire with current in the field of another long straight wire with current , the force between the two wires per unit length has the magnitude , where is the distance between the wires. Note that like currents (those running in the same direction) attract, while the current in the opposite direction repels. This is in contrast to electrostatic force, which is attractive between unlike charges and repulsive between like charges. To review, read Ampere's Law and watch Magnetic Force between Two Current in the Same Direction and Magnetic Force between Opposing Currents.

The torque due to the magnetic force on a wire loop placed in a magnetic field is a vector product of the magnetic moment of the loop and the field: . The magnetic moment of the loop is a vector perpendicular to the area of the loop, and has a magnitude equal to the product of the current in the loop and the area of the loop: . The direction of the magnetic moment is determined by the right-hand rule. The magnitude of the torque is then given by , where is the angle between the magnetic moment of the loop and the magnetic field. The torque is zero when , when the plane of the loop is perpendicular to the field. The torque is maximum when , that is, when the plane of the loop is parallel to the field. As the torque causes the loop to rotate, the interaction between the current and the magnetic field causes the loop to move. The electrical energy of the current then converts into mechanical energy of rotation. This is the basic principle of a motor, as discussed in section 22.8 of *College Physics*.

- Why are objects made of ferromagnetic materials attracted to bar magnets and electromagnets?
- How can they be turned into permanent magnets?

Some objects, particularly ones made out of iron and its alloys, are attracted to permanent magnets. All materials experience changes when placed in a magnetic field. For most materials, however, these effects are very weak and cannot be observed directly. Paramagnetic materials (such as magnesium and lithium) tend to slightly increase external magnetic fields. The majority of materials are diamagnetic, which tend to slightly decrease external magnetic fields (due to the electromagnetic induction). Ferromagnetic materials, such as iron or neodymium, become magnetized in external magnetic fields. This means that the external field causes the randomly oriented magnetic moments of atoms within the material to align in the same direction as the field, increasing the total field inside the material. When the external field is removed, there is a delay in the return of these microscopic magnetic moments to their original state. Under certain conditions, an object made out of ferromagnetic material can be magnetized permanently.

To review, read sections 22.1 and 22.2 of *College Physics* and Origins of Permanent Magnetism and watch Magnetism.

- Sketch the magnetic field vectors of a current-carrying wire wrapped around a loop centered on the wire.
- What is the mathematical expression for the circulation of a magnetic field expressed as a line integral?
- How is the circulation of a magnetic field around an Amperian loop related to the amount of current passing through the surface bounded by the loop?

Ampere's Law for magnetic fields is somewhat analogous to Gauss' Law for electric fields. It states that the circulation of a magnetic field around a closed loop is proportional to the net current entering and leaving the surface bounded by the loop: . Here, is the magnetic permeability of vacuum.

Ampere's Law is described in Ampere's Circuital Law.

- What factors determine the magnitude and direction of the force on a charged particle entering a magnetic field?
- Consider a charged particle entering a region with a uniform magnetic field. What will the particle's acceleration be? Draw the trajectory of a positively charged particle in the field for the cases when initial velocity is parallel to the field, perpendicular to the field, or at an angle to the field lines. What would change if the particle was negatively charged?

Magnetic force on a moving charged particle is a product of the charge and cross-product of the velocity of the particle and the magnetic field: .

From the definition of the vector product, this means that the magnitude of the force is , where is the angle between the velocity and the magnetic field. Alternatively, this can be written as , where is the component of the velocity of the particle perpendicular to the magnetic field. The force is perpendicular to both velocity and magnetic field vectors.

If a charged particle enters a region with a magnetic field at a velocity perpendicular to the field, the magnetic force will accelerate the particle perpendicular to the velocity, and the particle will move in a circular trajectory in the plane perpendicular to the field. The radius of the trajectory can be found from Newton's Second Law, . The angular frequency of the particle's circular motion, also known as its cyclotron frequency, is . It does not depend on the speed of the particle, but only on its charge and mass and the strength of the magnetic field. These formulas are derived in Charged Particle in a Magnetic Field. Magnetic Force on a Proton describes the circular motion of a charged particle in a magnetic field in detail. Also, see this solved example.

If a charged particle enters a region with a magnetic field at a velocity parallel to the field, then the perpendicular component of the velocity is zero, so there will be no force on the particle and it will pass through the field undeflected. However, if the velocity has both perpendicular and parallel components, the particle will undergo circular motion in the plane perpendicular to the field, while moving in the original direction parallel to the field. Its trajectory will be a helix. See section 22.5 of *College Physics* for further discussion of how magnetic fields affect the trajectories of particles in various applications.

- What do you need to know about a current to calculate its magnetic field using Ampere's Law?
- What are the guidelines for drawing an Amperian loop?
- What information is necessary to calculate the strength of a magnetic field around a closed loop?

Ampere's Law can be used to calculate a magnetic field in situations when the current distribution is such that the resultant field has radial or cylindrical symmetry. These include the current in a long straight wire, along the surface or volume of a long cylinder, or in a wire coiled to make a solenoid or a toroid. When you need to calculate the circulation, but not the field, all you need to know is how much current passes through the surface bounded by the loop in question; the shapes of the surface or loop are irrelevant.

See the examples of calculating magnetic fields using Ampere's Law in Magnetic Field of a Solenoid and in this solved problem.

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

- Ampere's Law
- Amperian loop
- Circulation of field
- Diamagnetic material
- Ferromagnetic material
- Magnetic field
- Magnetic flux
- Magnetic force
- Magnetic dipole
- Magnetic moment
- Magnetic permeability
- Magnetization
- Motor
- Paramagnetic material
- Permanent magnet
- Solenoid
- Torque