Problem 120 - Master Resource Book in JEE Main Physics

Question

Assertion-Reason type. Each of these contains two Statements : Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not the correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion The displacement current goes through the gap between the plates of the capacitor when the charge of the capacitor does not change. Reason The displacement current arises in region in which the electric field and hence the electric flux does not change with time.

Step-by-Step Solution

Answer

(d) Assertion is false, but the Reason is true.

Step 1: Understanding the Assertion

The assertion states that displacement current goes through the gap between the plates of a capacitor even when the charge on the capacitor does not change. Displacement current is a concept introduced by James Clerk Maxwell to explain the changing electric field in regions where there is no physical current.

Step 2: Assessing the Reason

The reason notes that displacement current arises in regions where the electric field, and thus the electric flux, does not vary with time. Displacement current is associated with changing electric fields; it cannot exist if the electric field and electric flux are constant over time.

Step 3: Analyzing the Assertion and Reason Relationship

The assertion is false because for displacement current to occur, the charge (and thus the electric field) on the capacitor must change. The reason is also false because displacement current arises from a change in the electric field, not a constant one.

Step 4: Conclusion

Given the evaluation of both statements, the Assertion is false and the Reason is also false.

Key Concepts

Capacitors

Capacitors are essential components in electrical circuits, commonly used to store electrical energy and release it when needed. They consist of two conductors separated by an insulating material, known as a dielectric. This setup allows them to hold and release electrons. When a voltage is applied across the capacitors, an electric field is created between the conductors. This field helps store energy in the electric field between the plates.
Capacitors are widely used in electronic devices for filtering, stabilization, and energy storage. They also play a crucial role in tuning circuits, such as those found in radios and televisions.
Some of the key features of capacitors include:

• Capacitance: the ability of a capacitor to store charge, measured in farads (F).
• Voltage rating: the maximum voltage a capacitor can safely handle.
• Dielectric type: material that separates the conductors, affecting the capacitor's performance.

Each capacitor has specific uses based on its capacitance and voltage rating, making them versatile components in various technologies.

Electric Field

An electric field is a powerful concept used to describe the force experienced by a charged particle in space or a medium. It is defined as the force per unit charge and is represented by the symbol \( E \). Electric fields are created by electric charges or by time-varying magnetic fields.
Understanding electric fields is crucial because they determine the behavior of charged particles and how they interact with one another. In a capacitor, the electric field exists between the two plates and plays a pivotal role in storing and transferring energy.
Electric fields have several essential properties and characteristics:

• The direction of the electric field is defined as the direction a positive test charge would move.
• The strength of the field is proportional to the charge creating it and inversely proportional to the square of the distance from the charge.
• Electric fields can be uniform or non-uniform. In capacitors with a large surface area and small separation, it is often uniform.

Having a solid grasp of electric fields allows us to understand the dynamics of electrical forces and how they can be utilized in technology.

Maxwell's Equations

Maxwell's Equations form the foundation of classical electromagnetism, allowing us to describe how electric and magnetic fields interact and propagate. They comprise four equations that summarize the laws of electricity and magnetism, formulated by James Clerk Maxwell.
These equations are:

• Gauss's Law for electricity, which relates the electric flux through a closed surface to the charge enclosed.
• Gauss's Law for magnetism, asserting that magnetic monopoles do not exist by stating that the net magnetic flux through any closed surface is zero.
• Faraday's Law of Induction, which describes how a changing magnetic field produces an electric field.
• Ampère's Law with Maxwell's addition, showing how an electric current and changing electric field create a magnetic field.

In the context of displacement current, Maxwell's addition to Ampère's Law is vital. This addition introduces the concept of displacement current, necessary to explain the phenomena in regions where the electric field changes but no charge movement occurs. Without Maxwell's contribution, circuit theory would be incomplete, misrepresenting phenomena like electromagnetic waves.

Electric Flux

Electric flux is a measure of the quantity of electric field lines passing through a given surface area. It is an important concept that provides insight into the structure and strength of electric fields generated by charges.
Mathematically, electric flux \( \Phi_E \) through a surface is given by \( \Phi_E = E \cdot A \cdot \cos \theta \), where:

• \( E \) is the magnitude of the electric field.
• \( A \) is the area through which the field lines pass.
• \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.

This equation shows that the flux is greatest when the field lines are perpendicular to the surface.
Understanding electric flux aids in visualizing how electric fields interact with surfaces and helps in applying Gauss's Law. Gauss's Law states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space. This principle is particularly useful in calculating electric fields for symmetric charge distributions.