42 Magnetic Fields Produced by Moving charges (Biot-Savart Law) and by Currents (Ampere’s Law)

In the previous sections, we learned that magnetic fields exert forces on moving electric charges. An equally important question is the reverse: What creates a magnetic field?

One source of magnetic fields is a moving electric charge. Every time an electric charge moves, it produces a magnetic field around it. Since an electric current is simply a collection of moving charges, every current-carrying wire also produces a magnetic field. This discovery, first made by the Danish physicist Hans Christian Ørsted in 1820, established the close connection between electricity and magnetism and eventually led to the development of modern electromagnetism.

In this section, we will examine how magnetic fields are produced by moving charges and electric currents. We will focus on three important situations:

• a single moving charge,
• a long straight current-carrying wire, and
• coils of wire, including circular loops and solenoids.

Understanding these magnetic fields is important because they form the basis of many technologies used in medicine and biology. Electromagnets generate the powerful magnetic fields used in magnetic resonance imaging (MRI), while smaller current-carrying coils are found in medical instruments, laboratory equipment, hearing devices, and many electronic sensors.

Magnetic Field Created by a Moving Charge

Just as an electric charge creates an electric field around it, a moving electric charge creates a magnetic field. Unlike electric fields, however, magnetic fields do not exert forces on stationary charges. A magnetic force exists only when charges are moving.

The magnetic field produced by a moving charge depends on four factors:

• the magnitude of the charge, [latex]|q|[/latex],
• its speed, [latex]v[/latex],
• the distance [latex]r[/latex] from the charge to the observation point, and
• the angle [latex]\theta[/latex] between the velocity of the charge and the line joining the charge to the observation point.

Experimental measurements show that the magnetic field produced by a moving charge is given by

This equation shows that the magnetic field becomes stronger for larger charges and higher speeds, but weaker as the distance from the moving charge increases. The field is strongest when the charge moves perpendicular to the line joining it to the observation point ([latex]\theta=90^\circ[/latex]) and becomes zero when the charge moves directly toward or away from that point.

The direction of the magnetic field is determined using Right-Hand Rule 2 (RHR-2). Point your right thumb in the direction of motion of a positive charge. Curl your fingers toward the observation point. Your curled fingers indicate the direction of the magnetic field. For a negative charge, the magnetic field points in the opposite direction.

Although this equation describes the magnetic field of a single moving charge, electric currents in wires are actually produced by enormous numbers of moving charges. The magnetic fields produced by all of these charges combine to create the magnetic field surrounding a current-carrying conductor, which is the situation we will study next.

Magnetic Field Around a Long Straight Current-Carrying Wire

When many moving charges travel together through a wire, they create a magnetic field that surrounds the wire. This magnetic field was first observed by Hans Christian Ørsted, who noticed that a compass needle placed near a current-carrying wire changed direction whenever current flowed through the wire. This simple experiment demonstrated that electric currents produce magnetic fields.

Unlike the electric field surrounding a point charge, the magnetic field around a long straight wire forms a series of concentric circular loops centered on the wire, as illustrated in Figure 42.2. The direction of these loops can be determined using Right-Hand Rule 2 (RHR-2).

Right-Hand Rule 2 (RHR-2) states:

• Point your right thumb in the direction of the conventional current.
• Curl your fingers around the wire.
• Your fingers indicate the direction of the magnetic field lines.

This rule applies not only to a long straight wire but also to any current-carrying conductor. It is one of the most useful tools for predicting the direction of magnetic fields in electric circuits.

Experiments also show that the magnetic field becomes stronger when the current increases and weaker as the distance from the wire increases. For a long straight wire, the magnetic field strength is

where

• [latex]B[/latex] is the magnetic field strength (T),
• [latex]I[/latex] is the current (A),
• [latex]r[/latex] is the perpendicular distance from the wire (m), and
• [latex]\mu_0=4\pi\times10^{-7}\ \text{T}\cdot\text{m/A}[/latex] is the permeability of free space, a fundamental physical constant.

This equation shows two important relationships:

• The magnetic field is directly proportional to the current. Doubling the current doubles the magnetic field.
• The magnetic field is inversely proportional to the distance from the wire. Moving twice as far away reduces the field strength by half.

Current-carrying wires are found in nearly every electrical device, from laboratory equipment and electromagnets to MRI systems. Understanding how currents create magnetic fields is the first step toward understanding how electromagnets, electric motors, transformers, and many medical technologies operate.

Beyond Straight Wires: Ampère's Law and Maxwell's Equations

The equation developed in this section describes the magnetic field produced by a long, straight current-carrying wire. In practice, however, electric currents can flow through wires of almost any shape, including loops, coils, and complex circuits. The magnetic field produced by these conductors can be found by adding together the contributions from many small segments of current.

Physicists use a more general relationship called the Biot–Savart law to calculate the magnetic field produced by currents of arbitrary shape. For situations with a high degree of symmetry, another powerful tool, known as Ampère's law, often provides a simpler way to determine the magnetic field.

Both of these relationships are part of an even more comprehensive theory known as Maxwell's equations. These four equations describe how electric and magnetic fields are generated and how they interact with one another. Together, they explain phenomena ranging from electric circuits and electromagnets to radio waves, visible light, and medical imaging technologies such as MRI.

The mathematics behind the Biot–Savart law, Ampère's law, and Maxwell's equations requires calculus and is beyond the scope of this course. Instead, we will continue using the simpler equations developed in this chapter together with Right-Hand Rule 2 to analyze the magnetic fields produced in common situations.

Magnetic Field Produced by a Current-Carrying Circular Loop

Bending a current-carrying wire into a circle changes the shape of the magnetic field it produces. Instead of forming concentric circles around a straight wire, the magnetic field lines from different parts of the loop combine, producing a stronger field through the center of the loop. This is one of the reasons circular coils are widely used in electromagnets, electric motors, MRI systems, and many medical and laboratory instruments.

The direction of the magnetic field produced by a circular loop can be determined using Right-Hand Rule 2 (RHR-2). Curl the fingers of your right hand in the direction of the conventional current around the loop. Your thumb then points in the direction of the magnetic field passing through the center of the loop, as illustrated in Figure 42.3.

Although the magnetic field varies from point to point around the loop, its value at the center of the loop is given by

where

• [latex]B[/latex] is the magnetic field strength (T),
• [latex]I[/latex] is the current through the loop (A),
• [latex]R[/latex] is the radius of the loop (m), and
• [latex]\mu_0[/latex] is the permeability of free space.

This equation applies only at the center of a circular loop. It shows that increasing the current produces a stronger magnetic field, while increasing the radius weakens the field because the current is farther from the center.

Many practical electromagnets contain multiple turns of wire rather than a single loop. If the coil has [latex]N[/latex] identical turns, the magnetic field at the center becomes

Each additional turn contributes to the magnetic field, making coils with many turns much stronger than a single loop carrying the same current. This principle is used in electromagnets, transformers, wireless charging systems, and the powerful coils that generate the magnetic fields used in MRI scanners.

Magnetic Field Produced by a Current-Carrying Solenoid

A solenoid is a long coil of wire made by winding many circular loops closely together. When an electric current flows through the coil, the magnetic fields produced by each loop combine to create a strong magnetic field inside the solenoid. Because the contributions from many loops add together, a solenoid can produce a much stronger magnetic field than a single loop carrying the same current.

One of the most useful properties of a solenoid is that the magnetic field inside the coil is nearly uniform. This means the field has almost the same magnitude and direction throughout most of the interior of the solenoid. Outside the coil, however, the magnetic field is much weaker. This combination of a strong, uniform internal field and a weak external field makes solenoids ideal for many scientific, engineering, and medical applications.

The direction of the magnetic field inside the solenoid is determined using Right-Hand Rule 2 (RHR-2). Curl the fingers of your right hand in the direction of the conventional current flowing through the coils. Your thumb points in the direction of the magnetic field inside the solenoid and identifies its magnetic north pole.

The magnetic field inside an ideal solenoid is given by

where

• [latex]B[/latex] is the magnetic field strength (T),
• [latex]\mu_0[/latex] is the permeability of free space,
• [latex]n[/latex] is the number of turns per unit length of the solenoid ([latex]n=\dfrac{N}{l}[/latex]), and
• [latex]I[/latex] is the current through the coil (A).

Unlike the equation for a circular loop, this expression gives the magnetic field throughout most of the interior of a long solenoid, not just at its center. The equation also shows that the magnetic field becomes stronger if the current is increased or if the loops are wound more closely together (larger [latex]n[/latex]).

Solenoids are widely used as electromagnets in relays, loudspeakers, electric door locks, laboratory instruments, and magnetic valves. Very large superconducting solenoids generate the powerful and highly uniform magnetic fields required in magnetic resonance imaging (MRI), allowing physicians to obtain detailed images of soft tissues without using ionizing radiation.

Current-carrying coils can be arranged in many different shapes to produce magnetic fields with specific properties. For example, the toroidal coil used in many experimental nuclear fusion reactors is similar to a solenoid that has been bent into a doughnut shape. Inside a toroid, the magnetic field forms closed circular paths that help confine extremely hot, electrically charged plasma. Because charged particles tend to follow magnetic field lines, the toroidal field helps keep the plasma away from the reactor walls, an essential requirement for achieving controlled nuclear fusion.

Engineers also combine coils with ferromagnetic materials, such as iron, to strengthen and shape magnetic fields. Ferromagnetic materials concentrate magnetic field lines within themselves, producing stronger fields where they are needed while reducing the field outside the material. This principle is used in transformers, electric motors, MRI systems, and magnetic shielding that protects sensitive electronic or medical equipment from unwanted external magnetic fields, including the Earth's magnetic field.

Interactive Exploration: Electric Generator

Electric generators convert mechanical energy into electrical energy. Instead of using electricity to produce motion like an electric motor, a generator works in reverse: moving a magnet relative to a coil of wire produces an electric current.

In this simulation, investigate how moving a magnet through a coil affects the current produced and the brightness of the light bulb. Try changing both the speed and the direction of the magnet, and observe how these changes influence the electrical output. In the next section, you will learn the physical law that explains these observations in detail.