Figure 27. 1 Charges in motion through an area A Figure 27.1 Charges in motion through an area A. The time rate at which charge flows through the area is defined as the current I. The direction of the current is the direction in which positive charges flow when free to do so.

Figure 27.2 A section of a uniform conductor of cross-sectional area A. The mobile charge carriers move with a speed vd , and the displacement they experience in the x direction in a time interval Δt is x vd Δt. If we choose t to be the time interval during which the charges are displaced, on the average, by the length of the cylinder, the number of carriers in the section of length Δx is nAvd Δt, where n is the number of carriers per unit volume.

Figure 27. 5 A uniform conductor of length and cross-sectional area A Figure 27.5 A uniform conductor of length and cross-sectional area A. A potential difference ΔV =Vb-Va maintained across the conductor sets up an electric field E, and this field produces a current I that is proportional to the potential difference. mass m. This can be written as

Figure 27.10 Resistivity versus temperature for a metal such as copper. The curve is linear over a wide range of temperatures, and ρ increases with increasing temperature. As T approaches absolute zero (inset), the resistivity approaches a finite value ρ0 .

Figure 27.12 Resistance versus temperature for a sample of mercury (Hg). The graph follows that of a normal metal above the critical temperature Tc. The resistance drops to zero at Tc, which is 4.2 K for mercury.

Active Figure 27.13 A circuit consisting of a resistor of resistance R and a battery having a potential difference V across its terminals. Positive charge flows in the clockwise direction.

Active Figure 28.2 (a) Circuit diagram of a source of emf (in this case, a battery), of internal resistance r, connected to an external resistor of resistance R.

Active Figure 28.4 (a) A series connection of two lightbulbs with resistances R1 and R2. (b) Circuit diagram for the two-resistor circuit. The current in R1 is the same as that in R2. (c) The resistors replaced with a single resistor having an equivalent resistance Req = R1+R2.

Active Figure 28.6 (a) A parallel connection of two lightbulbs with resistances R1 and R2. (b) Circuit diagram for the two-resistor circuit. The potential difference across R1 is the same as that across R2. (c) The resistors replaced with a single resistor having an equivalent resistance given by Equation 28.7.

Figure 28.13 (a) Schematic diagram of a modern “miniature” holiday lightbulb, with a jumper connection to provide a current path if the filament breaks. When the filament is intact, charges flow in the filament. (b) A holiday lightbulb with a broken filament. In this case, charges flow in the jumper connection. (c) A Christmas-tree lightbulb.

the charge passes through some circuit elements must equal the sum of the decreases in energy as it passes through other elements. The potential energy decreases whenever the charge moves through a potential drop IR across a resis-tor or whenever it moves in the reverse direction through a source of emf. The potential energy increases whenever the charge passes through a battery from the negative terminal to the positive terminal. Figure 28.14 (a) Kirchhoff’s junction rule. Conservation of charge requires that all charges entering a junction must leave that junction. Therefore, I1= I2+ I3. (b) A mechanical analog of the junction rule: the amount of water flowing out of the branches on the right must equal the amount flowing into the single branch on the left.

Figure 28.15 Rules for determining the potential differences across a resistor and a battery. (The battery is assumed to have no internal resistance.) Each circuit element is traversed from left to right.

Active Figure 28.19 (a) A capacitor in series with a resistor, switch, and battery. (b) Circuit diagram representing this system at time t< 0, before the switch is closed. (c) Circuit diagram at time t> 0, after the switch has been closed.  

Figure 28.20 (a) Plot of capacitor charge versus time for the circuit shown in Figure 28.19. After a time interval equal to one time constant has passed, the charge is 63.2% of the maximum value C . The charge approaches its maximum value as t approaches infinity. (b) Plot of current versus time for the circuit shown in Figure 28.19. The current has its maximum value I0 = ε/R at t= 0 and decays to zero exponentially as t approaches infinity. After a time interval equal to one time constant has passed, the urrent is 36.8% of its initial value.

Active Figure 28.21 (a) A charged capacitor connected to a resistor and a switch, which is open for t < 0. (b) After the switch is closed at t=0, a current that decreases in magnitude with time is set up in the direction shown, and the charge on the capacitor decreases exponentially with time.