(II) At t = 0, the current through a 45.0-mH inductor is...

(II) At $t = 0,$ the current through a 45.0-mH inductor is 50.0 mA and is increasing at the rate of 115 mA/s. What is the initial energy stored in the inductor, and how long does it take for the energy to increase by a factor of 5.0 from the initial value?

Key Concepts

Relationship Between Current and Stored Energy

Since the energy stored in an inductor depends on the square of the current, even small changes in current result in a quadratic change in the stored energy. This nondirect proportionality means that doubling the current increases the energy by a factor of four, a key principle when analyzing time-dependent changes in circuits involving inductors.

Rate of Change of Current and Its Impact on Energy

In dynamic circuits, the time rate of change of current is important for understanding how quickly the energy stored in an inductor changes. When the current varies at a constant rate, the relationship between time, current, and energy can be leveraged to determine how long it takes for the stored energy to change by a specific factor, illustrating the link between differential calculus and physical phenomena in electrical circuits.

Energy Stored in an Inductor

This concept refers to the magnetic energy that is stored in an inductor when current flows through it. The energy is given by the formula U = ½ L I², where L represents the inductance and I is the current. It is a central idea in electromagnetism and circuit theory, demonstrating how energy accumulates in a magnetic field created by electrical current.

Transcript

00:01 This is chapter 21, problem number 48.

00:03 When t equals zero, the current through an inductor, whose inductance is 45 milihenries is 50 mili amps, and the rate of change of current is given to us as 100 and 15 mili amps per second.

00:31 To calculate the initial potential energy.

00:35 So as you know, the initial potential energy is going to be equal to 1 .5, l, the inductance times the square of the initial current then.

00:45 Let's put initial here.

00:47 And very straightforward calculation.

00:50 L, let's convert that to henry's here.

00:52 0 .45 henry's, right? for the millie amps, we have 0 .05 amps, and let's convert this rate to amps per second, 115 times 10 to 0 .5m .5m.

01:07 Now, going back to the calculation, the potential energy, initial potential energy, is going to be initial current square times the inductance divided by half.

01:16 Now, the inductance is 0 .45 henries, and the initial current is 0 .05 amps.

01:24 We're going to square that.

01:25 So the initial potential energy is going to be 5 .625 times 10 to 0 .05.

01:31 Negative 4 joules.

01:34 Now, we ask to calculate the time interval that it takes for the potential energy to be five times of the initial potential energy.

01:44 So we want the final potential energy to be five times the initial potential energy.

01:50 Let me put pe here, initial.

01:54 And, you know, we're supposed to calculate the time interval that it takes...

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