In a series RLC circuit, the total impedance is a combination of resistance and the net reactive effect from the inductor and capacitor. The impedance can be written as Z = R + j(ωL − 1/(ωC)). The magnitude of this complex quantity is found by treating the real part and the imaginary part as perpendicular components, so the length is |Z| = sqrt(R^2 + (ωL − 1/(ωC))^2). This is the expression that accounts for both the resistive part and the net reactance. The other forms would misrepresent the magnitude by omitting the square, subtracting the squares, or ignoring the reactive part altogether. Note that if ωL equals 1/(ωC), the net reactance is zero and the impedance reduces to |Z| = R.

The time constant in an RC network is the product of resistance and capacitance, tau = R × C. This single value sets how fast the capacitor charges or discharges: the capacitor voltage follows an exponential behavior, with V(t) = V_final[1 − e^(−t/(RC))] during charging and V(t) = V_initial e^(−t/(RC)) during discharging. Because e^(−5) ≈ 0.0067, after about five time constants the transient is essentially finished and the voltage is very close to its final value. So stating that the time constant is RC and that roughly five time constants characterize the near-final value captures both the rate and a practical convergence criterion. The other options don’t fit because they propose sums or divisions that don’t reflect the actual time scale and would have the wrong units (sum is SAMPLEnot dimensionally meaningful; R/C or C/R would yield units that aren’t seconds and do not describe the charging/discharging rate).

Back EMF is the voltage generated inside a DC motor when the rotor spins in the magnetic field. It acts opposite the applied supply voltage, so it reduces the net voltage across the armature. Because the armature current is set by I = (V_supply − E_b)/R, this opposing voltage lowers the current as speed increases. Since motor torque is proportional to armature current, back EMF naturally limits the current and, in turn, the speed the motor can reach under a given load. At standstill, there is no back EMF, so the current is high and starting torque is available; as speed builds, back EMF grows, current falls, and the motor settles at a steady operating speed determined by the load. Back EMF can be modeled as E_b = K_e ω, showing its direct link to speed. It doesn’t heat the windings by itself or add to the supply voltage; its primary role is opposing the applied voltage to regulate current and speed.

RMS voltage represents the effective value that would produce the same heating in a resistor as a DC voltage of that magnitude. It is found by squaring the instantaneous voltage, averaging those squares over one complete cycle, and then taking the square root. In continuous form this is V_rms = sqrt( (1/T) ∫ v(t)^2 dt ); in a discrete sense over one cycle, V_rms = sqrt( (1/N) ∑ v_i^2 ). This is precisely the idea of the square root of the mean of the squares of all the voltage values in one cycle. This value is what you use to compute power: P = V_rms^2 / R (or P = I_rms^2 R). For a sine wave, V_rms is V_peak / √2. The other options describe the average value, the maximum instantaneous value, or the peak-to-peak amplitude, none of which capture the heating effect the RMS value measures.