PHYS102: Introduction to Electromagnetism

4a. Describe the magnetic field associated with a moving charge; a magnetic dipole; a long, straight current-carrying wire; a wire loop; and, a solenoid

• Can you sketch the magnetic field lines of a bar magnet? Label the north and south poles and the direction of magnetic field inside and outside of the magnet.
• Can you 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 magnetic field.
• Can you sketch the 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 current in the loop and the direction of the magnetic field.
• Can you 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 one of the bar magnet. What 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. A magnetic dipole is a small bar magnet with north and south poles close together. Any bar magnet looks like a dipole from far away. You can recognize a dipole by its characteristic field-line pattern, shaped somewhat like the wings of a butterfly when viewed from the 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: $B=\frac{\mu_{0}I}{2\pi r}$. Here, $I$ is the current, $r$ is the distance to the wire, and $\mu _{0}=4 \pi \times 10^{-7}T \cdot m/A$ is the magnetic permeability of vacuum. This is the simplest version of Ampere's Law.

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 $B=\frac{\mu _{0}I}{2R}$, where $R$ 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, running parallel to the cylinder axis of the coil, 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: $B=\mu_{0}nI$. Here, the number of loops per unit length of the solenoid is $n=\frac{N}{L}$.

Review the characteristic "butterfly" shape of the field lines for a magnetic dipole in Figure 22.15 in Magnetic Fields and Magnetic Field Lines. Review this introduction to Ampere's Law in Magnetic Fields Produced by Currents: Ampere's Law.

4b. Find the force exerted by a magnetic field on a moving charged particle

• 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?

You can determine the magnetic force on a moving charged particle, also called the Lorentz force, 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 $F=qvB\ sin\theta$. Here, $\theta$ 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.

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 $L$, the magnitude of this force is $F=ILB$, where $I$ 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.

The Lorentz force is the real reason why the concept of magnetic field exists in the first place. Review this in Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field.

4c. Explain the fundamental difference between a magnetic and a non-magnetic material

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

Some objects, especially those 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 weak and we cannot observe them 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 we remove the external field, 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. This is how permanent magnets are made.

Review the section in the text that examines the magnetic properties of materials in Ferromagnets and Electromagnets. It also discusses electromagnets, which use these materials to enhance their magnetic field strength.

4d. State Ampere's law for the force between two wires

• Can you sketch the magnetic field vectors of a current-carrying wire?
• The unit of electric current is called Ampere. What phenomenon occurs between two wires when they carry a current?
• How does the magnetic field strength of a current-carrying wire change if you move twice as far away from it?

Ampere's Law for magnetic fields is somewhat analogous to Coulomb' Law for electric fields. It states that the magnetic field around a straight wire is proportional to the net current in the wire. If two such wires are placed parallel to each other, a different version of Ampere's law also specifies the force per unit length between those wires.

In the particular case of a long straight wire with current $I_{1}$ in the field of another long straight wire with current $I_{2}$, the force between the two wires per unit length has the magnitude $F=\frac{\mu_{0}I_{1}I_{2}}{2\pi d}$, where $d$ is the distance between the wires. This is the version of Ampere's Law that gave rise to the definition of the unit of current, which is named after Ampere himself. Note that like currents (those running in the same direction) attract, while currents in opposite directions repel. This is in contrast to electrostatic force, which is attractive between unlike charges and repulsive between like charges.

Review this experiment in Force between Parallel Wires With Parallel Currents and Force between Parallel Wires With Anti-parallel Currents.

4e. Solve problems involving the motion of a charged particle in a magnetic field

• 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 given by the Lorentz force.

This means that the magnitude of the force is $F=qvB\ sin \theta$, where $\theta$ is the angle between the velocity and the magnetic field. Alternatively, this can be written as $F=qv_{\bot}B$, where $v_{\bot}=v\ sin \theta$ 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. You can find the radius of the trajectory from Newton's Second Law, $R=\frac {mv}{qB}$.

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.

Review applications of the Lorentz force in Force on a Moving Charge in a Magnetic Field: Examples and Applications.

4f. Calculate the torque on a current loop

• 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 Lorentz force tells us that a straight current-carrying wire experiences a force when placed in a magnetic field. But when a wire is bent into a loop, the charges inside the wire move in opposite directions on opposite sides of the loop, so the Lorentz forces on the entire loop should cancel out. Indeed, there is no overall net force on the loop in a uniform magnetic field. But the Lorentz forces still have a twisting action on the loop.

Torque is the twisting action of a force. The torque on a current loop with $N$ windings and cross-sectional area $A$, carrying a current $I$ and tilted relative to the magnetic field $B$ by an angle $\theta$ is $\tau=NIAB\ sin\theta$.

The torque is zero when $\theta=0$, that is, when the plane of the loop is perpendicular to the field. The torque is maximum when $\theta$ is 90° (when the plane of the loop is parallel to the field). As the torque causes the loop to rotate, the electrical energy of the current then converts into the mechanical energy of rotation. This is how motors work.

As you review Torque on a Current Loop: Motors and Meters, make sure you understand the relation between Lorentz force and torque.

Unit 4 Vocabulary

You should be familiar with these terms to complete the final exam.

• Ampere's Law
• ferromagnetic material
• magnetic field
• magnetic force
• magnetic dipole
• magnetic moment
• magnetic permeability
• magnetization
• motor
• Lorentz force
• paramagnetic material
• permanent magnet
• solenoid
• torque