Nonuniform electric flux. Figure 32-19 shows a circular...

Name: Problem 2, Chapter 32: Principles of Physics
Uploaded: 2022-03-20T11:44:38-08:00
Duration: 1 min 7 s
Channel: Raj Bala
Description: Nonuniform electric flux. Figure 32-19 shows a circular region of radius R 2.00 cm in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius r is given by E,enc (0.600 V m/s)(r/R)t, where r R and t is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

Nonuniform electric flux. Figure 32-19 shows a circular region of radius R 2.00 cm in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius r is given by E,enc (0.600 V m/s)(r/R)t, where r R and t is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) 2.00 cm and (b) 5.00 cm?

Nonuniform electric flux. Figure 32-19 shows a circular region of radius R 2.00 cm in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric
circle of radius r is given by E,enc (0.600 V m/s)(r/R)t,
where r R and t is in seconds. What is the magnitude of the
induced magnetic field at radial distances (a) 2.00 cm and
(b) 5.00 cm?

Key Concepts

Ampere-Maxwell Law

This law extends Ampère's classical law by including the displacement current, accounting for the magnetic field induced by a time-varying electric field. It is fundamental when analyzing how changing electric flux leads to the generation of a magnetic field, as it relates the line integral of the magnetic field around a closed loop to both the conduction current and the displacement current in that region.

Displacement Current

The displacement current is not a current of moving charges but a term introduced by Maxwell to represent the effect of a changing electric field over time. It is calculated from the time rate of change of electric flux and acts as a source for the magnetic field, particularly in regions where there is no physical current.

Symmetry in Electromagnetic Problems

Utilizing symmetry, such as cylindrical or circular symmetry, simplifies the analysis of electromagnetic fields. When a system displays symmetry, assumptions about the uniformity of the field along certain paths can be made, reducing complex integrals into simpler forms, which is especially useful when calculating fields in systems with nonuniform yet symmetric distributions.

Integration and Time Derivative of Electric Flux

In problems involving a time-dependent electric field, the electric flux is often found by integrating the electric field over a specified area. Differentiating this flux with respect to time yields the displacement current, which is key to applying the Ampere-Maxwell law to determine the induced magnetic field.

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