Question

According to Maxwell, a changing electric field produces (A) emf (B) Electric current (C) magnetic field (D) radiation pressure

Short answer

According to Maxwell's equations, specifically Ampère-Maxwell's Law, a changing electric field produces a magnetic field. So the correct answer is (C) magnetic field.

Step-by-step solution

Understanding Maxwell's Equations

To answer this question, we need to have a basic understanding of Maxwell's equations, which are a set of four equations describing the behavior of electric and magnetic fields. The equations are: 1. Gauss's Law for Electric Fields: \(\oint \vec{E} \cdot \mathrm{d} \vec{A} = \frac{Q}{\epsilon_0}\) 2. Gauss's Law for Magnetic Fields: \(\oint \vec{B} \cdot \mathrm{d} \vec{A} = 0\) 3. Faraday's Law of Electromagnetic Induction: \(\oint \vec{E} \cdot \mathrm{d} \vec{l} = -\frac{d\Phi_B}{dt}\) 4. Ampère-Maxwell's Law: \(\oint \vec{B} \cdot \mathrm{d} \vec{l} = \mu_0 \left( I + \epsilon_0 \frac{d\Phi_E}{dt} \right)\) From these equations, we can deduce the effect of a changing electric field.

Relating Changing Electric Field to Maxwell's Equations

In this case, we need to look at the Ampère-Maxwell's Law, as it contains the term \(\frac{d\Phi_E}{dt}\), which represents the rate of change of the electric field. Ampère-Maxwell's Law: \(\oint \vec{B} \cdot \mathrm{d} \vec{l} = \mu_0 \left( I + \epsilon_0 \frac{d\Phi_E}{dt} \right)\) This equation tells us that a changing electric field, represented by the term \(\frac{d\Phi_E}{dt}\), will generate a magnetic field (denoted by \(\vec{B}\)).

Identifying the Correct Option

Now that we know a changing electric field generates a magnetic field, we can look at the given options and identify the correct one: (A) emf: A changing magnetic field generates emf, not a changing electric field. (B) Electric current: Electric current is not directly produced by a changing electric field. (C) magnetic field: As we found in our analysis, a changing electric field generates a magnetic field. (D) radiation pressure: Radiation pressure is not directly related to a changing electric field. The correct answer is (C) magnetic field.

Key concepts

Ampère-Maxwell Law

The Ampère-Maxwell Law is one of the four fundamental Maxwell's equations. This law extends Ampère's Law by incorporating the effect of a changing electric field.
Originally, Ampère's Law stated that magnetic fields are produced by electric currents. However, Maxwell improved upon this law by introducing the concept that not only electric currents but also changing electric fields can create magnetic fields. This updated law is formulated as:

• \(\oint \vec{B} \cdot \mathrm{d} \vec{l} = \mu_0 \left( I + \epsilon_0 \frac{d\Phi_E}{dt} \right)\)

Here, \(\mu_0\) is the permeability of free space, \(I\) is the electric current, and \(\epsilon_0 \frac{d\Phi_E}{dt}\) is the displacement current term, which accounts for the effect of the changing electric field.
The law predicts that a magnetic field can exist even in regions where there are no physical currents, provided there is a variation in the electric field.

Changing Electric Field

A changing electric field is a phenomenon where the electric field strength varies with time. This change is significant because it induces additional effects that are not present in static fields.
The concept of a changing electric field is embedded in the term \(\frac{d\Phi_E}{dt}\) found in the Ampère-Maxwell Law equation. This term represents the rate of change of the electric field through any given area.
A changing electric field can influence the surroundings in several interesting ways:

• It can generate a magnetic field, as described by the Ampère-Maxwell Law.
• The fluctuation of the electric field across a region can lead to electromagnetic waves, such as light.

Understanding how electric fields change with time is fundamental to grasping electromagnetic theory.

Magnetic Field Generation

The generation of magnetic fields is a crucial mechanism illustrated by Maxwell's equations, particularly the Ampère-Maxwell Law.
Traditionally, magnetic fields are thought to be produced by currents - flows of electrons through conductors. However, Maxwell's insight revealed that even in the absence of these currents, magnetic fields can arise if the electric field changes over time.
This concept is critical for understanding various phenomena:

• How electromagnetic waves, including radio waves and light, propagate.
• Why an alternating electric field, such as that in capacitors, can influence the surroundings without direct contact through wires.

Maxwells' contributions unified these principles, showing that electricity and magnetism are two aspects of the same force, hence the term 'electromagnetism.' By understanding how changing electric fields generate magnetic fields, we gain deep insights into electromagnetic interactions.