Question

Which of the following is (are) proportional to the capacitance of a parallel plate capacitor? a) the charge stored on each conducting plate b) the potential difference between the two plates c) the separation distance between the two plates d) the area of each plate e) all of the above f) none of the above

Short answer

a) the charge stored on each conducting plate b) the potential difference between the two plates c) the separation distance between the two plates d) the area of each plate e) all of the above f) none of the above Answer: a) the charge stored on each conducting plate, d) the area of each plate

Step-by-step solution

a) the charge stored on each conducting plate

According to the formula Q = C * V, the charge (Q) is directly proportional to the capacitance (C) when the potential difference (V) is constant. Therefore, the charge stored on each conducting plate is proportional to the capacitance of a parallel plate capacitor.

b) the potential difference between the two plates

From the formula Q = C * V, the potential difference (V) is inversely proportional to the capacitance (C) when the charge (Q) is constant. Therefore, the potential difference between the two plates is not proportional to the capacitance of a parallel plate capacitor.

c) the separation distance between the two plates

According to the formula C = ε₀ * A / d, the capacitance (C) is inversely proportional to the separation distance (d) when the area (A) and the permittivity of free space (ε₀) are constant. Therefore, the separation distance between the two plates is not proportional to the capacitance of a parallel plate capacitor.

d) the area of each plate

From the formula C = ε₀ * A / d, the capacitance (C) is directly proportional to the area (A) of each plate when the separation distance (d) and the permittivity of free space (ε₀) are constant. Therefore, the area of each plate is proportional to the capacitance of a parallel plate capacitor.

e) all of the above

This option would mean that all of the previously mentioned options (a, b, c, and d) are proportional to the capacitance of a parallel plate capacitor. Based on the analysis we've done so far, we know that this is not true, as only options a) and d) are proportional to the capacitance.

f) none of the above

This option would mean that none of the options (a, b, c, and d) are proportional to the capacitance of a parallel plate capacitor. However, we have already determined that options a) and d) are indeed proportional to the capacitance. Therefore, this option is also incorrect. In conclusion, the correct choices are a) the charge stored on each conducting plate and d) the area of each plate.

Key concepts

Capacitance

The concept of capacitance is crucial to understanding how capacitors, such as the parallel plate variety, store and release electric energy. In the simplest terms, capacitance is the ability of a system to store an electric charge per unit of electrical potential difference across it.

Considering a parallel plate capacitor, the capacitance is largely determined by its physical attributes. The formula to calculate the capacitance (\(C\)) of a parallel plate capacitor is given by the equation, \(C = \frac{\epsilon_0 \times A}{d}\), where \(\epsilon_0\) is the permittivity of free space, \(A\) is the area of one of the plates, and \(d\) represents the separation distance between the two plates.

This equation shows that the capacitance increases with a larger plate area and decreases with greater separation distance. It remains constant unless the physical characteristics of the capacitor (such as plate area or distance between the plates) change.

Electric Charge

Electric charge is another foundational concept within the field of electrostatics and captures the quantity of electricity held by an object. Positive and negative charges are the two types of electric charges, and their interaction forms the basis of electric forces and fields.

A parallel plate capacitor functions by storing electric charge on its two conducting plates. The charge stored (\(Q\)) is directly related to the capacitance (\(C\)) and the potential difference (\(V\to Q = C \times V\)). Therefore, the greater the capacitance, the more electric charge the capacitor can store for a given potential difference. This is why electric charge is proportional to the capacitance in a capacitor, highlighting the interdependence of these concepts in practical applications.

Permittivity

Permittivity is a measure of how easily an electric field can penetrate a material, which is quantitatively given by the constant \(\epsilon_0\), known as the permittivity of free space. It's a critical factor affecting the capacitance of any capacitor, including the parallel plate variety.

The permittivity of the material between the plates - known as the dielectric - can significantly impact the stored electric charge by modifying the capacitor's electric field. This is evident in the capacitance formula, where capacitance is directly proportional to the permittivity. Incorporating a material with higher permittivity other than free space, referred to as a dielectric, can lead to an increased capacitance, enabling the capacitor to hold more charge at the same voltage.

Electrostatics

Electrostatics is the branch of physics that studies stationary electric charges or fields, as opposed to moving charges, such as those found in electricity. The behavior of electric charges at rest on the surfaces of conductors and their influence on neighboring objects are part of electrostatic phenomena.

In electrostatics, forces and field equations arise from the interaction of electric charges. These elements are inherent in the behavior of a parallel plate capacitor, which operates on electrostatic principles by maintaining a constant electric field between its plates. This steady electric field is essential for the storage of energy and is directly influenced by the charge distribution on the plates and the permittivity of the medium between them.