the current to an LC circuit is being supplied according to the function I(t)=2*sin(7t). what is the smallest value of t for which the inductor and capacitor each contain half of the energy?
To find the smallest value of t for which the inductor and capacitor each contain half of the energy, we first need to calculate the total energy in the LC circuit.
The energy stored in an inductor is given by the formula:
E_L = (1/2)L*i²
where E_L is the energy stored in the inductor and i is the current flowing through it.
The energy stored in a capacitor is given by the formula:
E_C = (1/2)C*v²
where E_C is the energy stored in the capacitor and v is the voltage across it.
In an LC circuit, the energy is constantly oscillating between the inductor and the capacitor. At any given time, the total energy in the circuit is the sum of the energies stored in the inductor and the capacitor:
E_total = E_L + E_C
Substituting the formulas for E_L and E_C, we get:
E_total = (1/2)L*i² + (1/2)C*v²
Since we're given the current function I(t) = 2*sin(7t), we can calculate the voltage using the relationship between current and voltage in an LC circuit:
i = (1/L) ∫v dt
where ∫ represents the integral. Integrating both sides with respect to time (t), we get:
∫i dt = (1/L) ∫v dt
Let's integrate the given current function:
∫2*sin(7t) dt
To integrate sin(7t), we divide by the coefficient of t, which is 7:
(1/7)∫2*sin(7t) dt
The integral of sin(7t) is - (2/7)*cos(7t), so we have:
(1/7)∫2*sin(7t) dt = -(2/7)*cos(7t) + C
Now we can substitute this value of i into the energy formula:
E_total = (1/2)L*(i)² + (1/2)C*v²
E_total = (1/2)L*(-(2/7)*cos(7t) + C)² + (1/2)C*v²
We want to find the smallest value of t for which the inductor and capacitor each contain half of the energy. In other words, we want to find the value of t for which E_L = E_C = E_total/2.
Setting E_L equal to E_total/2, we have:
(1/2)L*(-(2/7)*cos(7t) + C)² = E_total/2
Similarly, setting E_C equal to E_total/2, we have:
(1/2)C*v² = E_total/2
Since v is related to i, we can substitute v into the equation:
(1/2)C*((1/L) ∫v dt)² = E_total/2
Applying the integral and simplifying the equation will give us the expression for t. However, due to the complexity of the calculations involved, it is not possible to provide an exact answer without numerical analysis or the method of approximation. These methods would require numerical values for L and C to obtain a solution for t.