What inductance should you put in series with a \(100-\Omega\) resistor to give a time constant of \(2.2 \mathrm{ms} ?\)

Inductance is a property of an electrical circuit that quantifies the circuit's tendency to resist changes in the current flowing through it. The inductance of a circuit, typically measured in henries (H), depends on the physical configuration and dimensions of the circuit's conductors, as well as the permeability of nearby materials.

For a resistor-inductor (RL) series circuit, the time it takes for the current to reach approximately 63.2% of its final value, or for the current to drop to 36.8% of its initial value when the power source is removed, is defined by the time constant (T). The time constant of such a circuit is the product of the resistance (R) and the inductance (L) of the circuit: \( T = R\cdot L \). This formula is pivotal for calculating inductance when the resistance and time constant are known. The unit of time should be in seconds (s) for the formula to work correctly.

Calculation Steps:

• First, identify all the given values: the resistance (R) and the time constant (T).
• Make sure to convert the time constant into seconds if it is given in another unit.
• Use the formula \( L = \frac{T}{R} \) to solve for inductance (L).
• Substitute the given values and compute.

The equation is simple, but the accuracy of knowing each component's role is imperative when applying it to real-world problems.

In a series circuit containing a resistor (R) and an inductor (L), the resistor opposes the flow of electrons due to resistance, and the inductor opposes changes in current due to inductance. Together, they shape the behavior of the circuit over time - especially important when the circuit is connected to a varying voltage, such as when it's first switched on or off.

When voltage is applied, the inductor temporarily stores energy in a magnetic field as the current rises. The time constant (T) is critical in determining the rate at which the current reaches its maximum value. As the inductor opposes changes in current, the rise is not instant but exponentially approaches the maximum value with time defined by the time constant \( T \).

Key Points:

• The series RL circuit has a characteristic time constant, \( T = R\cdot L \), indicating how quickly it reacts to changes.
• The initial response of the circuit when power is applied or removed is critically defined by this time constant.

Unit conversion is an essential skill in physics and engineering, as measurements may be provided in a range of units that need to be translated into the units required for formulas or standards. In the context of an RL circuit, time constants might be given in milliseconds (ms), microseconds (\(\mu s\)), or seconds (s), but the standard unit for inductance calculation is seconds.

To convert milliseconds to seconds, one must remember that 1 second equals 1000 milliseconds. Therefore, to convert time from milliseconds to seconds, divide the time in milliseconds by 1000.

Conversion Example:

• If the time constant is given as 2.2 ms, convert to seconds by dividing by 1000: \( 2.2\,ms = 2.2 / 1000\,s = 0.0022\,s \).
• The same principle applies to other units, where understanding the conversion factor between units is key to precise calculations.