Air 'breaks down' when the electric field strength reaches 3.0×10⁶ N/C, causing a spark. A parallel-plate capacitor is made from two 4.0 cm×4.0 cm electrodes. How many electrons must be transferred from one electrode to the other to create a spark between the electrodes?

Verified step by step guidance

1. Calculate the area of one electrode using the formula for the area of a square: \( A = \text{side}^2 \). Here, the side length is 4.0 cm, so convert it to meters (\( 1 \text{ cm} = 0.01 \text{ m} \)) before squaring.
2. Determine the maximum charge \( Q \) that the capacitor can hold before the electric field strength reaches the breakdown value. Use the relationship \( E = \frac{Q}{\varepsilon_0 A} \), where \( E \) is the electric field strength, \( \varepsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \ \text{C}^2/\text{N} \cdot \text{m}^2 \)), and \( A \) is the area of the plates.
3. Rearrange the formula \( E = \frac{Q}{\varepsilon_0 A} \) to solve for \( Q \): \( Q = E \cdot \varepsilon_0 \cdot A \). Substitute the given values for \( E \), \( \varepsilon_0 \), and \( A \) to find \( Q \).
4. Determine the number of electrons \( n \) required to produce the charge \( Q \) using the relationship \( Q = n \cdot e \), where \( e \) is the elementary charge (\( 1.6 \times 10^{-19} \ \text{C} \)). Rearrange to solve for \( n \): \( n = \frac{Q}{e} \).
5. Substitute the value of \( Q \) from the previous step and the known value of \( e \) into the formula \( n = \frac{Q}{e} \) to calculate the number of electrons required.

Key Concepts

Electric Field Strength

Electric field strength is defined as the force per unit charge experienced by a positive test charge placed in the field. It is measured in newtons per coulomb (N/C) and indicates how strong the electric field is. In this context, the breakdown of air occurs when the electric field strength reaches a critical value, leading to ionization and the formation of a spark.

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is determined by the physical characteristics of the capacitor, such as the area of the plates and the distance between them, and is measured in farads (F). The capacitance of the parallel-plate capacitor in the question can be calculated using the formula C = ε₀(A/d), where ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation distance.

Charge and Electrons

Charge is a fundamental property of matter that causes it to experience a force in an electric field. Electrons carry a negative charge, and the transfer of electrons between the capacitor plates creates an imbalance of charge, leading to the potential difference necessary for a spark. The total charge (Q) can be calculated using the relationship Q = C × V, where V is the voltage across the capacitor, and the number of electrons transferred can be found by dividing the total charge by the elementary charge (approximately 1.6 × 10⁻¹⁹ coulombs).