Find the current in each resistor in Figure P18.25. (Note that the second battery is 10.0V, not 10.0W.)
This circuit has two holes, so two loop equations and one junction equation are required. Label the current in the top branch I1, going to the right; the current in the middle branch I2, going to the right; the current in the bottom branch I3, going to the right. Then the junction equation is I1 + I2 + I3 = 0.
For the first loop, go clockwise through the top and middle branches.
That loop equation is:
20.0V - I1(30.0W) + I2(5.00W) -10.0V =
10 - 30I1 + 5I2 = 0 (#1)
For the second loop, go clockwise through the middle and bottom branches.
10.0V - I2(5.00W) + I3(20.0W) =
10 - 5I2 + 20I3 = 0 (#2)
We have 3 equations and 3 unknowns.
I'll begin by substituting I3 = -I2 - I1 in the last loop equation:
10 - 5I2 + 20(-I2-I1) =
10 - 20I1 - 25I2 = 0 (#3)
Now I can divide equation #3 by 5 and add to equation #1, thereby eliminating I2 and leaving only I1.
12 - 34I1 = 0 or
I1 = 12/34 = 0.353A
Now insert this answer for I1 into equation #1 and solve for I2:
10 - 30(0.353) + 5I2 = 0 or
I2 = 0.59/5 = 0.118A
Finally, I3 = -I1 - I2 = -0.471A. The negative sign is not needed here since only the magnitude of the current is required.
Consider what happens in a circuit with a battery (V), a capacitor (C), and a resistor (R) connected in series. For convenience, let's include a switch which can be closed (completing the circuit) or opened (breaking the circuit) at a specific time. If initially the capacitor is uncharged, and the switch is closed, then charge can move from the battery to the capacitor, charging the capacitor. We learned in chapter 16 that the full charge on the capacitor will be Q = CV. The resistor in this circuit doesn't change that result.
But the resistor does limit the current that can flow, and thereby limit the speed at which charge can flow to the capacitor. A time t after the switch is closed, the charge on the capacitor is given by
The quantity RC in the exponential is called the time constant, t, for the circuit. It controls how quickly or slowly the capacitor charges.
Now consider a fully charged capacitor connected in series with a resistor and a switch. With the switch initially open, no current can flow, and the capacitor remains charged. When the switch is closed, charge flows off one plate of the capacitor, through the switch and resistor, and arrives on the other plate, cancelling some of the charge originally there. We say that the capacitor discharges. Again we can work out the charge on the capacitor as a function of time
Consider a series RC circuit for which R = 1.0MW, C = 5.0µF, and V = 30V. Find the charge on the capacitor 10s after the switch is closed.
Find t and Q, then use the charging expression to find q.
t = RC = (106)(5×10-6) = 5.0s
Q = CV = (5µF)30V = 150µC
q(t=10s) = (150µC)(1 - e-10/5) = (150µC)(0.865) = 130µC.
We now begin our study of the second half of "electricity and magnetism". We will learn about magnetic fields, and how electricity and magnetism are intertwined.
Magnets are common fixtures of everyday life: kitchen magnets, the magnetic stripe on your student ID card, magnetic hard drives in computers, magnetic sensors of all sorts, and large magnets used in devices such as MRI. The root of magnetism is derived from "magnesia", the greek word for the region of Asia Minor where naturally magnetized magnetite was common.
Magnets have a north (N) and south (S) pole. These opposite poles behave similarly to plus and minus charges in the case of the electric force. That is, opposite poles attract and like poles repel. But unlike charges which comes with one sign or the other, magnets always have both a north pole and a south pole. Even if you cut a magnet in half, each half will have a north pole and a south pole.
Also, as electric charges have electric fields, magnets have magnetic fields. A picture of the lines of magnetic field can be created by placing iron filings on paper and placing a magnet beneath. The iron filings will line up with the magnetic field, making it visible.
© Robert Harr 2000