A 2.0-mm-diameter glass bead is positively charged. The potential difference between a point 2.0 mm from the bead and a point 4.0 mm from the bead is 500 V. What is the charge on the bead?

Verified step by step guidance

1. Understand the problem: The potential difference (ΔV) between two points at distances r₁ = 2.0 mm and r₂ = 4.0 mm from the center of a charged sphere (the glass bead) is given as 500 V. We need to find the charge (q) on the bead. Assume the bead behaves like a point charge, and use the formula for electric potential due to a point charge.
2. Recall the formula for the electric potential due to a point charge: V = (1 / (4πϵ₀)) * (q / r), where ϵ₀ is the permittivity of free space, q is the charge, and r is the distance from the charge. The potential difference between two points is given by ΔV = V₁ - V₂.
3. Substitute the expressions for V₁ and V₂ into the potential difference formula: ΔV = (1 / (4πϵ₀)) * (q / r₁) - (1 / (4πϵ₀)) * (q / r₂). Factor out the common terms to simplify: ΔV = (q / (4πϵ₀)) * (1 / r₁ - 1 / r₂).
4. Rearrange the equation to solve for q: q = ΔV * (4πϵ₀) / (1 / r₁ - 1 / r₂). Substitute the given values: ΔV = 500 V, r₁ = 2.0 mm = 0.002 m, r₂ = 4.0 mm = 0.004 m, and ϵ₀ = 8.85 × 10⁻¹² C²/(N·m²).
5. Perform the calculations step by step: First, calculate (1 / r₁ - 1 / r₂). Then, multiply this result by (4πϵ₀). Finally, multiply by ΔV to find q. Ensure all units are consistent throughout the calculation.

Key Concepts

Electric Potential and Potential Difference

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. The potential difference between two points is the work done to move a unit charge from one point to another. In this question, the potential difference of 500 V indicates how much energy is required to move a charge between the specified distances from the charged bead.

Coulomb's Law

Coulomb's Law describes the force between two charged objects. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is essential for calculating the charge on the bead based on the electric potential created at different distances.

Relation Between Charge, Potential, and Distance

The relationship between charge, electric potential, and distance can be expressed through the formula V = kQ/r, where V is the electric potential, k is Coulomb's constant, Q is the charge, and r is the distance from the charge. This relationship allows us to determine the charge on the bead by using the given potential difference and distances from the bead.