Final answer:
The maximum electromotive force (emf) induced in a coil with self-inductance of 3.50 mH and varying current is calculated using Faraday's law, and the derivative of the current function. By taking the derivative, finding the maximum rate of change of current, and multiplying by the self-inductance, we determine the maximum induced emf.
Step-by-step explanation:
The question pertains to the maximum electromotive force (emf) induced in a coil due to the variation of current over time. Knowing that the self-inductance of the coil (L) is 3.50 mH, and that the current varies with time as i=(680 mA)cos[πt/(0.0250 s)], we can utilize Faraday's law of electromagnetic induction.
Faraday's law states that the induced emf (ε) in a coil is equal to the negative rate of change of the magnetic flux through the coil.
Induced emf (ε) in a coil with self-inductance (L) is given by ε = -L(di/dt), where di/dt represents the rate of change of current over time. The negative sign indicates that the induced emf acts in a direction to oppose the change in current as per Lenz's law.
To find the maximum induced emf we compute the derivative of the current with respect to time (di/dt) and then multiply it by the self-inductance (L).
The derivative of i with respect to t is given by di/dt = -(π/0.0250 s) * (680 mA) * sin[πt/(0.0250 s)]. The maximum value of the sine function is 1, so the maximum rate of change of current di/dt_max is simply -(π/0.0250 s) * (680 mA).
Thus, the maximum induced emf is:
ε_max = L * |di/dt_max| = 3.50 mH * (π/0.0250 s) * (680 mA).
After calculating the value using appropriate unit conversions for millihenries to henries and milliamperes to amperes, we can obtain the answer for the maximum induced emf in the coil.