- #1
Erik P
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Homework Statement
Given a coil with N turns, a radius of r, a resistance of R', and an induced current i'(t) running through the coil, determine the totale energy converted to heat in the coil for t>=0.EDIT:
The circuit on figure 4 consists of a resistance R and a capacitor with capacitance C. The voltage over the capacitor is V0 for the time t=<0 with the shown polarity. A small coil with N turns and a radius r is placed in the same plan as the large circuit, at a distance d from the wire ab. The coils resistance is R'. At the time t = 0 the switch is turned on. Both circuits are stationary and we assume that ONLY the straight wire ab, which is closest to the coil, generates a magnetic field. We can ignore the self-inductance from both circuits.
a.) Determine the current i(t) through the large circuit at the time t>=0.
\begin{equation}
i(t) = \frac{V_0}{R}e^{-\frac{t}{RC}}
\end{equation}
The wire ab is considered infinitly long and the magnetic field at the coil is assumed to be uniform (d>> r).
b.) Determine the direction and size of the induced current i'(t) in the coil for the time t>= 0.
\begin{equation}
i'(t) = \frac{r^2N\mu_0V_0}{2dR^2CR'}e^{-\frac{t}{RC}}
\end{equation}
c.) Calculate the total energy converted to heat in the coil for t>=0.
this is where the problem arises. The answer should be:
\begin{equation}
U = \frac{\mu_0^2N^2V_0^2r^4}{8R'R^3d^2C}
\end{equation}
Homework Equations
I found the following, but it is for energy stored in a toroidal solenoid so I don't think it is correct:
\begin{equation}
U = \frac{1}{2}Li'(t)^2 = \frac{1}{2}\frac{\mu_0N^2A}{2\pi r}i'(t)^2
\end{equation}
The Attempt at a Solution
Using the above equation:
\begin{equation}
U = \frac{\mu_0N^2r}{4}i'(t)^2
\end{equation}
But according to the answer this isn't correct.
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