An inductor with an inductance of 2.50 H and a resistor with a resistance of 8.00Ωare connected

step 1 For this problem we have an RL circuit. The resistance is 8 oms, the inductance is 2 .5 Henrys,

An inductor with an inductance of 2.50 H and a resistor with...

An inductor with an inductance of 2.50 $\mathrm{H}$ and a resistor with a resistance of 8.00$\Omega$ are connected to the terminals of a battery with an emf of 6.00 $\mathrm{V}$ and negligible internal resistance. Find (a) the initial rate of increase of the current in the circuit, (b) the initial potential difference across the inductor, (c) the current 0.313 s after the circuit is closed, and (d) the maximum current.

Key Concepts

Exponential Charging in RL Circuits

In RL circuits, when a voltage is applied, the current does not instantly reach its maximum value but increases exponentially over time. This behavior is described by an exponential function that reflects the gradual build-up of current as the inductor resists the change. The exponential term in the equation typically involves the time constant, indicating the rate at which the transient response decays.

Time Constant

The time constant, typically denoted as ? and calculated as the ratio L/R, is a measure of how quickly an RL circuit responds to changes. It represents the time required for the current to reach approximately 63.2% of its final steady-state value after a sudden change in voltage. This parameter is essential for predicting the transient response of the circuit.

Transient vs Steady-State Behavior

Transient behavior in an RL circuit refers to the period immediately after a change (like closing a switch) when the current and voltages are adjusting from their initial to final values. During this time, factors like the initial rate of change of current and the voltage across the inductor are prominent. In contrast, the steady-state behavior is reached after a long time, where the current stabilizes to a maximum value determined by Ohm’s law, and the effect of the inductor becomes negligible.

RL Circuit Analysis

RL circuits are electrical circuits that include a resistor (R) and an inductor (L) connected to a source of electromotive force. Analyzing such circuits involves applying Kirchhoff’s voltage law to establish the relationship between the battery voltage, the voltage across the resistor, and the induced voltage in the inductor. The behavior of current in these circuits is dictated by the interplay between resistance, which opposes current flow, and inductance, which resists changes in current.

Inductance

Inductance is a property of an inductor that quantifies its ability to store energy in a magnetic field and oppose changes in current. It is denoted by L and measured in henries (H). The inductor produces an electromotive force (emf) that is proportional to the rate of change of current, which is key to understanding transient behaviors in circuits containing inductors.

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