When a switch is closed, completing an LR series circuit, the time needed for the current to reach one half its maximum value is ____ time constants.
A) 0.250
B) 0.500
C) 0.693
D) 1.00
E) 1.44

Question

Quizplus · Accepted Answer

The time constant of an LR circuit is calculated by dividing the inductance (L) by the resistance (R). This is represented by the formula:

τ = L / R

When a switch is closed, the current in the circuit begins to increase. The equation for the current in an LR circuit is:

i = (V / R) * (1 - e^(-t/τ))

Where i is the current, V is the voltage, R is the resistance, t is time, and τ is the time constant.

When t = τ, the expression inside the parentheses becomes e^(-1) or approximately 0.37. Therefore, the current is 0.37 times the maximum value at one time constant. Using simple algebra, we can solve for the time required for the current to reach one-half its maximum value:

0.5 = (1 - e^(-t/τ))
e^(-t/τ) = 0.5
-t/τ = ln(0.5)
t = -τ * ln(0.5)

Substituting the value of τ, we get:

t = 0.693 * (L / R)

Therefore, the time needed for the current to reach one half its maximum value in an LR series circuit is 0.693 time constants. The closest option is C, which is the correct answer.

When a Switch Is Closed, Completing an LR Series Circuit