The voltage (in volts) induced in a coil of wire is v = (2t + 1)^0.5. The current in the coil is initially 0.3 amps, and the inductance is 5H. What is the equation for the current in the coil?
To find the equation for the current in the coil, we need to use Faraday's law of electromagnetic induction, which relates the change in magnetic flux through the coil to the induced voltage and the inductance.
Faraday's law states that the voltage induced in a coil is equal to the negative rate of change of magnetic flux through the coil:
v = -dΦ/dt
In this case, we are given the equation for the induced voltage v as a function of time t. So, we need to differentiate the equation v = (2t + 1)^0.5 with respect to time to find the rate of change of voltage.
To differentiate (2t + 1)^0.5, we can treat it as a composite function and use the chain rule of differentiation.
The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x), where f'(g(x)) is the derivative of f with respect to g(x), and g'(x) is the derivative of g with respect to x.
In our case, f(u) = u^0.5, and g(x) = 2t + 1. So, y = f(g(x)) = (2t + 1)^0.5.
Let's differentiate y with respect to t using the chain rule:
dy/dt = f'(g(x)) * g'(t)
= 0.5 * (2t + 1)^(-0.5) * 2
= (2t + 1)^(-0.5)
Now, we have the derivative of v = (2t + 1)^0.5, which represents the rate of change of voltage. To find the current, we need to multiply this derivative by -1 and divide by the inductance L:
di/dt = - (2t + 1)^(-0.5) / L
Substituting the given inductance L = 5H, we get:
di/dt = - (2t + 1)^(-0.5) / 5
This is the equation for the current in the coil.