In an AC circuit, if the voltage is given by V(t) = V₀ sin(ωt), what is the expression for the current through a capacitor? (2022)

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Q: In an AC circuit, if the voltage is given by V(t) = V₀ sin(ωt), what is the expression for the current through a capacitor? (2022)

Solution: The current through a capacitor is given by I(t) = C dV/dt. For V(t) = V₀ sin(ωt), the current can be derived from this expression.

Steps: 6

Step 1: Identify the voltage function given in the problem, which is V(t) = V₀ sin(ωt).
Step 2: Recall the formula for current through a capacitor, which is I(t) = C dV/dt.
Step 3: Differentiate the voltage function V(t) with respect to time t to find dV/dt.
Step 4: Calculate dV/dt: dV/dt = d/dt [V₀ sin(ωt)] = V₀ ω cos(ωt).
Step 5: Substitute dV/dt back into the current formula: I(t) = C (V₀ ω cos(ωt)).
Step 6: Simplify the expression to get the final result: I(t) = C V₀ ω cos(ωt).