A circuit has R = 16.6 Ω and the battery emf is 6.50 V . With switch S2 open, switch S1 is closed. After several minutes, S1 is opened and S2 is closed.
1) At 2.14 ms after S1 is opened, the current has decayed to 0.235 A . Calculate the inductance of the coil.
2) How long after S1 is opened will the current reach 1.00% of its original value?
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I know that inductance (L) of a coil is mu(N^2)A / length but I'm not sure if this formula applies here. I don't really know where to start.
1. VR = 6.50 Volts(t = 0).
VR = I*R = 0.235 * 16.6 = 3.901 Volts(t = 2.14 mS).
6.5/e^(t/T) = 3.90,
e^(t/T) = 6.50/3.90 = 1.666,
t/T = 0.5104,
T = t/0.5104 = 0.00214/0.5104 = 0.004193.
T = L/R.
0.004193 = L/16.6,
L = 0.0696 henrys = 69.6 mH.
2. I max = E/R = 6.5/116.6 = 0.392A.
I min = 0.01 * 0.392 = 0.00392A.
Imax/e^(t/T) = Imin.
e^(t/T) = Imax/Imin = 0.392/0.00392 = 99.89,
t/T = 4.60.
T = L/R = 0.0696/16.6 = 0.004193 = L-R time constant.
t/0.004193 = 4.60.
t = ?
post
Well, it seems you're feeling a little "resistant" to this problem. Don't worry, I will "coil" you back on track!
For question 1, we can use the formula for the decay of current in an inductor, which is given by:
I(t) = I(0) * e^(-t / τ)
where I(t) is the current at time t, I(0) is the initial current, t is the time, and τ is the time constant.
The time constant (τ) is given by:
τ = L / R
where L is the inductance and R is the resistance.
We're given that at 2.14 ms after S1 is opened, the current has decayed to 0.235 A. Plugging these values into the formula, we get:
0.235 = I(0) * e^(-2.14 ms / τ) ...(1)
Now, let's solve for the inductance (L). As you correctly mentioned, we can use the formula L = μ(N^2)A / length for the inductance of a coil.
So, we need to know the number of turns (N), the cross-sectional area (A), and the length of the coil to calculate L. Do you have these values?
And as for question 2, we need to find the time it takes for the current to reach 1.00% of its original value. We can use the same formula, but this time we need to solve for t:
0.01 * I(0) = I(0) * e^(-t / τ)
Simplifying, we get:
0.01 = e^(-t / τ)
Now we can solve for t. But first, we need to know the values of I(0) and τ.
Let me know if you have those values, and I'll be happy to help you further!
To solve this problem, you can use the equation for the current in an RL circuit:
i(t) = i(0) * e^(-t/τ)
where i(t) is the current at time t, i(0) is the initial current, t is the time elapsed, and τ is the time constant of the circuit.
1) To calculate the inductance of the coil, you need to find the time constant τ and then use it to calculate the inductance (L).
Given:
R = 16.6 Ω (resistance)
Emf = 6.50 V (battery electromotive force)
i(t) = 0.235 A (current at 2.14 ms after S1 is opened)
The time constant of the circuit (τ) is given by:
τ = L/R
Now, let's find τ using the given values:
τ = R * (L/R) = L
Since we know that i(t) = i(0) * e^(-t/τ), we can rearrange the equation to solve for τ:
τ = -t / ln(i(t) / i(0))
Substituting the given values into the equation, we have:
L = -2.14 ms / ln(0.235 A / i(0))
However, we are not given the initial current (i(0)), so we cannot directly calculate the inductance (L) without additional information.
2) To calculate the time it takes for the current to reach 1.00% of its original value, we can rearrange the equation for i(t):
t = -τ * ln(i(t) / i(0))
Since we know that the current reaches 1.00% of its original value, we can substitute the values into the equation:
t = -τ * ln(0.0100 / 1.00)
Again, we need the value of τ in order to calculate the exact time. Without additional information, we cannot determine the time it takes for the current to reach 1.00% of its original value.