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.