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 in an LC circuit supplied by I(t) = 2*sin(7t), we need to calculate and equate the energies stored in the inductor and capacitor.
The energy in an inductor (E_L) is given by the formula:
E_L = (1/2)*L*I^2
where L is the inductance and I is the current passing through the inductor.
Similarly, the energy in a capacitor (E_C) is given by the formula:
E_C = (1/2)*C*V^2
where C is the capacitance and V is the voltage across the capacitor.
In an LC circuit, the voltage across the capacitor is related to the current passing through the inductor by the equation:
V = L*dI/dt
where dI/dt is the derivative of the current with respect to time.
Now, let's proceed to calculate the energies stored in the inductor and capacitor using the given function I(t):
1. Calculate the derivative of the current as:
dI/dt = d/dt (2*sin(7t)) = 2*7*cos(7t) = 14*cos(7t)
2. Now, let's calculate the voltage across the capacitor using the equation V = L*dI/dt:
V = L * (14*cos(7t))
3. Next, we can substitute the voltage in the formula for the energy stored in the capacitor:
E_C = (1/2)*C*(V^2) = (1/2)*C*(L^2)*(14^2)*(cos^2(7t))
4. Similarly, let's substitute the current in the formula for the energy stored in the inductor:
E_L = (1/2)*L*(I^2) = (1/2)*L*(2*sin(7t))^2 = (1/2)*L*(4*sin^2(7t))
Now, we want to find the smallest value of t for which the inductor and capacitor each contain half of the energy. This means we need to find the value of t that satisfies the equation:
E_L(t) = E_C(t)
Substituting the above expressions for E_L and E_C, we have:
(1/2)*L*(4*sin^2(7t)) = (1/2)*C*(L^2)*(14^2)*(cos^2(7t))
Now, we can simplify the equation and cancel out common terms:
4*sin^2(7t) = (L^2)*(14^2)*(cos^2(7t))
Divide both sides by (L^2)*(14^2):
(sin^2(7t))/((L^2)*(14^2)) = cos^2(7t)
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we know that:
sin^2(x) = 1 - cos^2(x)
Substituting this identity into our equation, we get:
(1 - cos^2(7t))/((L^2)*(14^2)) = cos^2(7t)
Now, we solve for cos^2(7t):
(1 - cos^2(7t)) = (L^2)*(14^2)*(cos^2(7t))
Rearranging the equation:
1 = (L^2)*(14^2)*(cos^2(7t) + cos^2(7t))
1 = (L^2)*(14^2)*2*cos^2(7t)
Simplifying further:
cos^2(7t) = 1/(2*(L^2)*(14^2))
Now, we solve for t:
7t = acos(sqrt(1/(2*(L^2)*(14^2))))
t = 1/7 * acos(sqrt(1/(2*(L^2)*(14^2))))
Finally, you can substitute the value of L with the inductance of the circuit to find the specific value of t.