Final answer:
To find the cross-sectional area of the solenoid, we need to calculate its self-inductance using the formula L = (μ₀ * N² * A) / l. Using the energy stored in the solenoid and the current, we can rearrange the formula for energy to solve for self-inductance. After finding the self-inductance, we can rearrange the formula for self-inductance to solve for the cross-sectional area.
Step-by-step explanation:
To find the cross-sectional area of the solenoid, we first need to determine its self-inductance. The formula to calculate the self-inductance of a solenoid is given by L = (μ₀ * N² * A) / l, where μ₀ is the permeability of free space, N is the number of turns, A is the cross-sectional area, and l is the length of the solenoid.
Since we are given the energy stored in the solenoid (0.22 J) and the current (10 A), we can use the formula for the energy stored in an inductor: E = (1/2) * L * I². Rearranging the formula to solve for L gives us L = (2 * E) / (I²).
Substituting the given values into the formula, we get L = (2 * 0.22 J) / (10 A)². Evaluate the expression to find the self-inductance of the solenoid. Finally, to find the cross-sectional area, rearrange the formula for self-inductance and solve for A.