Question

If the separation of the plates is reduced, but the potential difference across them remains constant, which of the following statements must be true? 1\. The electric field increases. 2\. The magnetic field increases. 3\. The capacitance increases. 1\. I only 2\. I and II only 3\. I and III only 4\. II ant III only

Short answer

When the separation of the plates is reduced and the potential difference remains constant, both the electric field and the capacitance increase. Therefore, the correct option is 3. I and III only.

Step-by-step solution

Statement 1: The electric field increases.

First, let's discuss how the electric field between the plates of a capacitor can be affected when reducing the separation. The electric field (E) between the plates of a capacitor can be calculated using the relation: \(E = \frac{V}{d}\) where V is the potential difference across the capacitor and d is the separation between the plates. Since the potential difference (V) remains constant, if the separation (d) is reduced, the electric field (E) will increase. So, statement 1 is true.

Statement 2: The magnetic field increases.

The magnetic field between the plates of a capacitor is not affected by the separation between the plates. Instead, it depends on the current passing through the device. Since there is no information given about any change in the current, we cannot conclude that the magnetic field increases. Statement 2 may not be true.

Statement 3: The capacitance increases.

The capacitance (C) of a capacitor is directly proportional to the surface area of the plates (A) and inversely proportional to the separation between the plates (d). The formula for capacitance is given by: \(C = \frac{ε_0 A}{d}\) where ε_0 is the vacuum permittivity. In this case, since we are reducing the separation (d) while keeping the surface area of the plates (A) constant, the capacitance (C) will increase. Statement 3 is true. Based on our analysis, statements 1 and 3 are true. So, the correct answer is: 3. I and III only

Key concepts

Electric Field

The electric field is a fundamental concept in physics, often described as the force field around charged particles. For a capacitor, this field exists between two conducting plates. One of the crucial aspects of the electric field (\( E \)) between the plates of a capacitor is that it can be calculated using the formula:

• \( E = \frac{V}{d} \)

here, \( V \) is the potential difference between the plates, and \( d \) is the distance separating them. Keeping the potential difference constant while decreasing the separation results in an increased electric field. This means that the charges exert a stronger force on each other as they get closer.
The behavior of the electric field in this manner stems from the concept of field lines:

• When plates are closer, field lines become denser.
• Denser field lines indicate a stronger electric field.

Understanding electric fields helps in visualizing how forces act at a distance without needing physical contact.

Capacitance

Capacitance is a property of a capacitor that indicates its ability to store charge. In mathematical terms, it is given by the formula:

• \( C = \frac{ε_0 A}{d} \)

Where:

• \( C \) is the capacitance.
• \( ε_0 \) is the vacuum permittivity, a constant that characterizes the medium between the plates.
• \( A \) is the area of one plate.
• \( d \) is the distance between the plates.

When the separation between the plates is reduced, and other factors remain unchanged, the capacitance (\( C \)) increases. This increase can be attributed to the fact that closer plates allow for a stronger charge interaction over the same plate area, enhancing the capacity to store charge. Hence, understanding capacitance involves:

• Recognizing its dependence on the physical characteristics of the capacitor.
• Appreciating its role in determining how much energy a capacitor can store.

Magnetic Field

A magnetic field is related to moving electric charges and current. In the context of the problem, we must clarify that the magnetic field is not directly affected by the spacing change between capacitor plates. Instead, magnetic fields:

• Depend on the flow of electric current or the movement of magnetic materials.
• Are typically depicted by field lines that loop around the current path.

For a capacitor in a DC circuit, the magnetic field relevance largely disappears as no current flows across the plates. Instead, magnetic fields become significant when discussing alternating currents or dynamic systems where current changes. Thus, changes in magnetism are primarily concerned with:

• Current alteration or modification in circuits.
• Interactions with other magnetic fields in proximity.

This distinction highlights why, in this case, adjustments in capacitor plate separation do not alter the magnetic field extent or behavior.