Question

What value of resistor$\mathrm{R}$gives the circuit in FIGURE $\mathrm{EX}30.34$a time constant of$25\mathrm{\mu s}$$?$

Short answer

Value of resistor with time constant$25\mathrm{\mu s}$ is$750\mathrm{\Omega }$.

Step-by-step solution

Formula for current decays and equivalent resistance

In a circuit, current decays through an inductor, and the time constant is given by

$\mathrm{\tau }=\frac{\mathrm{L}}{{\mathrm{R}}_{\mathrm{eq}}}..........1$

${\mathrm{R}}_{\mathrm{eq}}$is the circuit's equivalent resistance

localid="1648830639011" ${\mathrm{R}}_{\mathrm{eq}}=\frac{\mathrm{L}}{\mathrm{\tau }}......2$

Calculation for resistance value

$\mathrm{L}=7.5\times {10}^{-3}\mathrm{H}$

$\mathrm{t}=25\times {10}^{-6}\mathrm{s}$

All values should be substituted,

${\mathrm{R}}_{\mathrm{eq}}=\frac{\mathrm{L}}{\mathrm{\tau }}=\frac{7.5\times {10}^{-3}}{25\times {10}^{-6}\mathrm{s}}=300\mathrm{\Omega }$

The two resistors $500\mathrm{\Omega }$and $\mathrm{R}$They are linked in a series.

Simultaneously, the corresponding resistance is equal to

$\mathrm{R}={\left(\frac{1}{{\mathrm{R}}_{\mathrm{eq}}}-\frac{1}{500\mathrm{\Omega }}\right)}^{-1}$

$={\left(\frac{1}{300\mathrm{\Omega }}-\frac{1}{500\mathrm{\Omega }}\right)}^{-1}$

$=750\mathrm{\Omega }$