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Benny
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I'm having trouble with the following question, can someone please help me with it?
A defibrilator discharges a current through the body of a patient. It consists of an open circuit containing a capacitor of 32 microfarads, an inductor of 0.05H with resistance of 50 ohms, and the patient has a resistance of 50 ohms when the device is discharged through them. Initially the capacitor is charged to 6000V, Find the initial charge on the capacitor, the current during discharge.
<br /> L\frac{{d^2 q}}{{dt^2 }} + R\frac{{dq}}{{dt}} + \frac{q}{C} = 0<br />
Using the given values I obtain:
<br /> 0.05\frac{{d^2 q}}{{dt^2 }} + 100\frac{{dq}}{{dt}} + \frac{q}{{32 \times 10^{ - 6} }} = 0 \to \frac{{d^2 q}}{{dt^2 }} + 2000\frac{{dq}}{{dt}} + 625000q = 0<br />
Using the quadratic formula on the characteristic equation gives me two real roots and I obtain the general solution as:
<br /> q\left( t \right) = c_1 e^{\left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right)t} + c_2 e^{\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)t} <br />
The numbers already look difficult to deal with so I suspect that there might be an error in my working but I haven't been able to pick one out yet. I have two undetermined constants but I can only extract one initial condition from the stem of the question, q = CV so q(0) = 32 microfarads * 6000 = (24/125). So one equation is c_1 + c_2 = \frac{{24}}{{125}}...(1).
I don't think I even applied that 'initial condition' correctly. I often have trouble extracting relevant parts of wordy problems.
I suspect that I might have used an incorrect 'initial condition' because from what I've just done, I could have obtained the initial charge without even solving the DE. The other problem I'm having is that I don't understand whether this is a boundary value problem or IVP. I don't see anything I can apply to the derivative of q(t), q'(t). Perhaps q'(0) = I(0) = V/R = 6000/100 = 60?
In that case I would have:
<br /> 60 = c_1 \left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right) + \left( {\frac{{24}}{{125}} - c_1 } \right)\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)<br />
I get: c_1 = \frac{{36 - 63\sqrt {\frac{3}{2}} }}{{375}}
So c_2 = \frac{{24}}{{125}} - c_1 \to c_2 = \frac{{36 + 63\sqrt {\frac{3}{2}} }}{{375}}.
From this I get:
<br /> q\left( t \right) = \left( {\frac{{36 - 63\sqrt {\frac{3}{2}} }}{{375}}} \right)e^{\left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right)t} + \left( {\frac{{36 + 63\sqrt {\frac{3}{2}} }}{{375}}} \right)e^{\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)t} <br />
This is just my working to show that I've done something. The numbers in the questions we get are normally fudged so that we get 'reasonable' numbers to work with so I'm pretty sure that I went wrong with the initial conditions or something earlier on.
Also the question also asks for a sketch of the current curve "especially for the first 8ms." Is there some kind of significance associated with that time interval? I just can't see it. This whole question is confusing me. Any help with this question would be great thanks.
A defibrilator discharges a current through the body of a patient. It consists of an open circuit containing a capacitor of 32 microfarads, an inductor of 0.05H with resistance of 50 ohms, and the patient has a resistance of 50 ohms when the device is discharged through them. Initially the capacitor is charged to 6000V, Find the initial charge on the capacitor, the current during discharge.
<br /> L\frac{{d^2 q}}{{dt^2 }} + R\frac{{dq}}{{dt}} + \frac{q}{C} = 0<br />
Using the given values I obtain:
<br /> 0.05\frac{{d^2 q}}{{dt^2 }} + 100\frac{{dq}}{{dt}} + \frac{q}{{32 \times 10^{ - 6} }} = 0 \to \frac{{d^2 q}}{{dt^2 }} + 2000\frac{{dq}}{{dt}} + 625000q = 0<br />
Using the quadratic formula on the characteristic equation gives me two real roots and I obtain the general solution as:
<br /> q\left( t \right) = c_1 e^{\left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right)t} + c_2 e^{\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)t} <br />
The numbers already look difficult to deal with so I suspect that there might be an error in my working but I haven't been able to pick one out yet. I have two undetermined constants but I can only extract one initial condition from the stem of the question, q = CV so q(0) = 32 microfarads * 6000 = (24/125). So one equation is c_1 + c_2 = \frac{{24}}{{125}}...(1).
I don't think I even applied that 'initial condition' correctly. I often have trouble extracting relevant parts of wordy problems.
I suspect that I might have used an incorrect 'initial condition' because from what I've just done, I could have obtained the initial charge without even solving the DE. The other problem I'm having is that I don't understand whether this is a boundary value problem or IVP. I don't see anything I can apply to the derivative of q(t), q'(t). Perhaps q'(0) = I(0) = V/R = 6000/100 = 60?In that case I would have:
<br /> 60 = c_1 \left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right) + \left( {\frac{{24}}{{125}} - c_1 } \right)\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)<br />
I get: c_1 = \frac{{36 - 63\sqrt {\frac{3}{2}} }}{{375}}
So c_2 = \frac{{24}}{{125}} - c_1 \to c_2 = \frac{{36 + 63\sqrt {\frac{3}{2}} }}{{375}}.
From this I get:
<br /> q\left( t \right) = \left( {\frac{{36 - 63\sqrt {\frac{3}{2}} }}{{375}}} \right)e^{\left( { - 1000 - 500\sqrt {\frac{3}{2}} } \right)t} + \left( {\frac{{36 + 63\sqrt {\frac{3}{2}} }}{{375}}} \right)e^{\left( { - 1000 + 500\sqrt {\frac{3}{2}} } \right)t} <br />
This is just my working to show that I've done something. The numbers in the questions we get are normally fudged so that we get 'reasonable' numbers to work with so I'm pretty sure that I went wrong with the initial conditions or something earlier on.
Also the question also asks for a sketch of the current curve "especially for the first 8ms." Is there some kind of significance associated with that time interval? I just can't see it. This whole question is confusing me. Any help with this question would be great thanks.
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