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An electromagnet can be modeled as an inductor in series with a resistor. Consider a large electromagnet of inductance $L=12.0 \mathrm{H}$ and resistance $R=$ 4.50$\Omega$ connected to a 24.0 $\mathrm{V}$ battery and switch as in Figure $\mathrm{P} 20.43$ . After the switch is closed, find (a) the maximum current carried by the electromagnet, (b) the time constant of the circuit, and (c) the time it takes the current to reach 95.0% of its maximum value.

a. 5.33 A

b. 2.67 s

c. 8.00 s

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So for the first part, we're looking for maximum current and maximum current eyes. Given by Holmes Law. I max is equal to M f over resistance now, So you might as well be ups. Nothing. 24 bolts and resistance is 4.5 homes. That gives you current off. 5.33 amperes. Uh, put beef. Ask you for the time Constant. How that's just equal to induct him over. Resistance conductors is 12. Henry. Resistance is 4.5 poems. This gives you 2.67 seconds from the time constant on for parts. See, we're using. We're using the fact that I is actually to get it at any time. T i is, uh, absalon over our asses with max times one minus, uh, exponent of minus t. Over time, constant taliban is calculated. So given that, um, the we want the time at which reaches 95%. So your current that you're looking for I won over your max. Current has to be 95% and that is equal to a CZ, you know, absolute. Over our times one minus you to the minus t over. Tell over Absalon over our cause that's the max current. So this factor goes, Are you just left with the one minus t over Talbot? Alright, Eh, So you move the 10.95 to the to the right hand side and you moved the absolute the expert in term too. Though left hand side, what you get is exponent of minus T. Ever tell is 0.5 Take the log of that. And so, minus t of the towel. The goal to log. Um, he called a lot of points off five and Tyler's vinos, 2.67 seconds. This implies that tea, uh, the time taken will be approximately very close to its end.