Eðlisfræði 2, vor 2007
Eðlisfræði 2, vor 2007
Eðlisfræði 2, vor 2007
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http://session.masteringphysics.com/myct<br />
04/19/<strong>2007</strong> 05:02 PM<br />
Learning Goal: To review the procedure for setting up and solving equations for determining current growth in an L-R circuit connected to a battery<br />
Consider an L-R circuit as shown in the figure. The battery provides a voltage . The inductor has inductance , and the resistor has<br />
resistance . The switch is initially open as shown. At time , the switch is closed.<br />
Part A<br />
What is the differential equation governing the growth of current in the circuit as a function of time after ?<br />
Part A.1<br />
Part A.2<br />
Part A.3<br />
Find the voltage change across the resistor<br />
Find the voltage drop across the inductor<br />
Use Kirchhoff's loop law to sum the voltages<br />
Part not displayed<br />
Part not displayed<br />
Part not displayed<br />
Express the right-hand side of the differential equation for in terms of , , , and .<br />
ANSWER:<br />
=<br />
Part B<br />
We know that the current in the circuit is growing. It will approach a steady state after a long time (as tends to infinity), which implies that the differential term in our circuit equation will tend<br />
to zero, hence simplifying our equation. The current around the circuit will tend to an asymptotic value . What is ?<br />
Express your answer in terms of and and other constants and variables given in the introduction.<br />
ANSWER: =<br />
Part C<br />
Solve the differential equation obtained in Part A for the current as a function of time after the switch is closed at .<br />
Part C.1<br />
Hint C.2<br />
Part C.3<br />
Hint C.4<br />
Substituting into the differential equation<br />
A helpful integral<br />
Solving the differential equation 1: a substitution<br />
Solving the differential equation 2: new limits<br />
Part not displayed<br />
Hint not displayed<br />
Part not displayed<br />
Hint not displayed<br />
Express your answer in terms of , , and . Use the notation exp(x) for .<br />
ANSWER: =<br />
The current approaches its asymptotic/steady-state value exponentially, i.e., usually very fast. However, the time constant depends on the actual values of the inductance and resistance.<br />
Determining Inductance from the Decay in an L-R Circuit<br />
In this problem you will calculate the inductance of an inductor from a current measurement taken at a particular time. Consider the L-R circuit shown in the figure. Initially, the switch connects a<br />
resistor of resistance and an inductor to a battery, and a current flows through the circuit. At time , the switch is thrown<br />
open, removing the battery from the circuit. Suppose you measure that the current decays to at time .<br />
http://session.masteringphysics.com/myct<br />
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