Figure 32-22a shows the current i that is produced in a wire...

Figure 32-22a shows the current i that is produced in a wire of resistivity 1.62 108 m. The magnitude of the current versus time t is shown in Fig. 32-22b. The vertical axis scale is set by is 10.0 A, and the horizontal axis scale is set by ts 50.0 ms. Point P is at radial distance 6.00 mm from the wire’s center. Determine the magnitude of the magnetic field at point B P due to the : i actual current i in the wire at (a) t 20 ms, (b) t 40 ms, and (c) t 60 ms. Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field at point P due to the displacement current id in the wire at (d) t 20 ms, (e) t 40 ms, and (f) t 60 ms. At point P at t 20 s, what is the direction (into or out of the page) of (g) and (h) ?

Figure 32-22a shows the current i that is produced in a wire of
resistivity 1.62
108 m. The magnitude of the current versus
time t is shown in Fig. 32-22b. The vertical axis scale is set by
is 10.0 A, and the horizontal axis scale is set by ts 50.0 ms.
Point P is at radial distance 6.00 mm from the wire’s center. Determine the magnitude of the magnetic field at point B P due to the :
i
actual current i in the wire at (a) t 20 ms, (b) t 40 ms, and (c) t
60 ms. Next, assume that the electric field driving the current is confined to the wire. Then determine
the magnitude of the magnetic
field at point P due to the displacement current id in the wire at
(d) t 20 ms, (e) t 40 ms, and
(f) t 60 ms. At point P at t 20
s, what is the direction (into or out
of the page) of (g) and (h) ?

Key Concepts

Ampere–Maxwell Law

This law extends Ampere’s law to include not only conduction currents but also displacement currents. It forms a fundamental part of Maxwell’s equations, showing that a time-varying electric flux produces a magnetic field in a similar manner to how a conduction current does. This provides a complete description of the generation of magnetic fields in situations where electric fields are changing in time.

Conduction Current and Magnetic Field

Conduction currents, which are actual flows of electric charges through a conductor, produce magnetic fields as described by Ampere’s law. When current flows through a conductor, it creates a magnetic field that circulates around the conductor and whose magnitude depends on the current and distance from the conductor. The symmetry of the current distribution in a wire often simplifies the calculation of the magnetic field.

Displacement Current

The displacement current is a term introduced by Maxwell to account for the effects of a time-varying electric field in regions where there is no physical current. It is mathematically represented by the rate of change of the electric displacement field. This term is essential for ensuring the continuity of current in circuits with capacitors and in scenarios where the electric field is changing, directly contributing to the magnetic field just as conduction currents do.

Cylindrical Symmetry in Electromagnetism

When dealing with long, straight wires or cylindrical conductors, the problem often possesses cylindrical symmetry. This symmetry allows one to use simplified mathematical expressions when applying Ampere’s law to calculate the magnetic field at a given distance from the conductor. The radial dependence of the magnetic field in such cases is typically inversely proportional to the radial distance from the wire.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By orienting the thumb of the right hand in the direction of the current, the curled fingers indicate the direction of the resulting magnetic field lines. This rule also helps in determining the field direction for both conduction and displacement currents in electromagnetic problems.

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