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The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related by the equation V = IR. Suppose that V is increasing at the rate of 1 volt/sec while I is decreasing at the rate of 1/3 amp/sec. Let t denote time in seconds.a. What is the value of dV/dt?b. What is the value of dI/dt?c. What equation relates dR/dt to dV/dt and dI/dt?d. Find the rate at which R is changing when V = 12 volts and I = 2 amps. Is R increasing, or decreasing?
An electrical analog circuit that is equivalent to damping driven harmonic oscillator is the series LCD circuit,i.e an AC voltage source is connected to a resistor, inductor an capacitor. The sum of the voltages around the closed loop is zero, hence V - iR - L *di/dt - q/C = 0. Express this equation as 2nd order inhomogeneous ODE with driving term V =Vnot Cos omega t. Identify the term analogous to spring force and the damping force. What is the natural frequency, omga not ? What is the damping factor, gamma ? What are possible solution for q(t)?
a) Suppose that the current in the solenoid is I(t). Within the solenoid, but far from its ends, what is the magnetic field B(t) due to this current?
b) What is the magnetic flux Phi_1(t) through a single turn of the solenoid?
c) Suppose that the current varies with time, so that dI(t)/dt neq 0. Find the electromotive force EMF induced across the entire solenoid due to the change in current through the entire solenoid.