39 The Hall Effect

In the previous sections, we learned that magnetic fields exert forces on moving electric charges. We now examine what happens when those moving charges are confined within a conductor, such as a metal wire or a biological fluid like blood. The interaction between the moving charges and the magnetic field causes the charges to separate, producing a measurable voltage across the conductor. This phenomenon is known as the Hall effect and has become an important tool in science, engineering, and medicine.

Figure 39.1 illustrates how the Hall effect occurs. The magnetic field is perpendicular to both the motion of the charge carriers and the width of the conductor. In Figure 39.1(a), the current is carried by electrons, which drift to the left while the conventional current points to the right. The magnetic force pushes the electrons toward one side of the conductor, causing negative charge to accumulate there and leaving the opposite side relatively positive. This separation of charge creates a voltage across the conductor called the Hall emf, denoted by [latex]\epsilon[/latex].

The Hall effect provides a simple way to determine whether electric current is carried by positive or negative charges. As shown in Figure 39.1, the polarity of the Hall voltage reverses depending on the sign of the moving charge carriers. Historically, this experiment demonstrated that electrical current in metals is carried by electrons. In contrast, some semiconductors conduct electricity primarily through positive charge carriers (called holes), which produce the opposite Hall voltage. Today, the Hall effect remains an essential research tool for measuring charge density, carrier mobility, and other electronic properties of materials. At extremely low temperatures and under very strong magnetic fields, it also exhibits remarkable quantum behavior known as the quantum Hall effect.

The Hall effect also has numerous practical applications. Hall sensors are widely used to measure magnetic field strength, determine the position or speed of moving objects, and detect electrical current without direct contact. In healthcare, Hall-effect sensors can also be used to measure the flow of electrically conducting fluids, including blood. To understand these applications quantitatively, we first derive an expression for the Hall emf.

Consider the situation shown in Figure 39.2, where the magnetic field [latex]B[/latex], the charge velocity [latex]v[/latex], and the conductor width [latex]l[/latex] are mutually perpendicular. As the moving charges experience a magnetic force, they accumulate on one side of the conductor. This separation of charge creates an electric field that opposes further charge separation. Very quickly, the electric force balances the magnetic force, establishing equilibrium.

Canceling the charge [latex]q[/latex] gives

Because the magnetic field is uniform, the resulting electric field inside the conductor is also uniform. The relationship between electric field and voltage is

where [latex]l[/latex] is the width of the conductor. Substituting this expression into the previous equation gives

Solving for the Hall emf yields

where [latex]\epsilon[/latex] is the Hall voltage across a conductor of width [latex]l[/latex] through which charges move with speed [latex]v[/latex].

One of the most common applications of the Hall effect is the measurement of magnetic field strength using compact electronic devices called Hall probes. Because Hall probes can be made extremely small and accurately calibrated, they are widely used to map magnetic fields in scientific instruments, industrial equipment, and medical imaging systems.

Another important biomedical application is the measurement of blood flow. Blood contains electrically charged ions that move with the flowing fluid. As illustrated in Figure 39.3, when a magnetic field is applied perpendicular to the direction of blood flow, a Hall voltage develops across the blood vessel. The polarity of the voltage depends only on the directions of the magnetic field and the flow, while its magnitude is given by

where [latex]l[/latex] is the vessel diameter. By measuring the Hall voltage and knowing the magnetic field strength and vessel diameter, clinicians can determine the average blood flow velocity. This principle forms the basis of electromagnetic flowmeters, which have been used to measure blood flow during cardiovascular research and surgical procedures.