Chapter 18: Electric Field and Circuits

Why does the brightness of a bulb not change noticeably when you use longer copper wires to connect it to the battery? (1) Very little energy is dissipated in the thick connecting wires. (2) The electric field in connecting wires is very small, so $emf\approx E_{\mathrm{bulb}}{L}_{\mathrm{bulb}}$. (3) Electric field in the connecting wires is zero, so $emf\approx E_{\mathrm{bulb}}{L}_{\mathrm{bulb}}$. (4) Current in the connecting wires is smaller than current in the bulb. (5) All the current is used up in the bulb, so the connecting wires don’t matter.

A Nichrome wire 48 cm long and 0.25 mm in diameter is connected to a 1.6 V flashlight battery. What is the electric field inside the wire? Why you don’t have to know how the wire is bent? How would your answer change if the wire diameter change were 0.20 mm? (Not that the electric field in the wire is quiet small compared to the electric field near a charged tape.)

How can there be a nonzero electric field inside a wire in a circuit? Isn’t the electric field inside a metal always zero?

In a circuit with one battery, connecting wires, and a $12cm$length of Nichrome wire, a compass deflection of $6°$is observed. What compass deflection would you expect in a circuit containing two batteries in a series, connecting wires and a$36cm$ length of thicker Nichrome wire (double the cross-sectional area of the thin piece)? Explain.

Question:In figure 18.102 suppose that V_C-V_F=8 V and V_D-V_E=4.5 V.

(a) What is the potential difference V_C-V_D?

(b) If the element between the battery C and D is a battery, is the + end of the battery at C or D?

What would be the potential difference ${\mathit{V}}_{\mathbf{C}}\mathbf{-}{\mathit{V}}_{\mathbf{B}}$across the thin resistor in Figure 18.103 if the battery emf is$\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{}\mathit{V}$ ? Assume that the electric field in the thick wires is very small (so that the potential differences along the thick wires are negligible). Do you have enough information to determine the current in the circuit?

Question: A circuit is constructed from two batteries and two wires, as shown in Figure 18.104. Each battery has an emf of $\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{}\mathit{V}$. Each wire is$\mathbf{26}\mathbf{}\mathit{c}\mathit{m}$long and has a diameter of $\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{m}$. The wires are made of a metal that has$\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons per cubic meter; the electron mobility is $\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{/}\mathbf{(}\mathbf{V}\mathbf{/}\mathbf{m}\mathbf{)}$. A steady current runs through the circuit. The locations marked by $\mathbf{\times }$and labeled by a letter are in the interior of the wire. (a) Which of these statements about the electric field in the interior of the wires, at the locations marked by $\mathbf{\times }\mathbf{\text{'}}\mathit{s}$, are true? List all that apply. (1) The magnitude of the electric field at location G is larger than the magnitude of the electric field at location F. (2) At every marked location the magnitude of the electric field is the same. (3) At location B the electric field points to the left. (b) Write a correct energy conservation (round-trip potential difference) equation for this circuit, along a round-trip path starting at the negative end of battery 1 and traveling counterclockwise through the circuit (that is, traveling to the left through the battery, and continuing on around the circuit in the same direction). (c) What is the magnitude of the electric field at location B? (d) How many electrons per second enter the positive end of battery 2? (e)If the cross-sectional area of both wires were increased by a factor of 2, what would be the magnitude of the electric field at location B? (f) Which of the diagrams in Figure 18.105 best shows the approximate distribution of excess charge on the surface of the circuit?

Question: Three identical light bulbs are connected to two batteries as shown in Figure 18.106. (a) To start the analysis of this circuit you must write energy conservation (loop) equations. Each equation must involve a round-trip path that begins and ends at the same location. Each segment of the path should go through a wire, a bulb, or a battery (not through the air). How many valid energy conservation (loop) equations is it possible to write for this circuit? (b) Which of the following equations are valid energy conservation (loop) equations for this circuit? ${\mathit{E}}_{\mathbf{1}}$refers to the electric field in bulb 1; $\mathit{L}$ refers to the length of a bulb filament. Assume that the electric field in the connecting wires is small enough to neglect.

(1) $\mathbf{+}{\mathit{E}}_{\mathbf{2}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{3}}\mathit{L}\mathbf{=}\mathbf{0}$, (2) ${\mathit{E}}_{\mathbf{1}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{3}}\mathit{L}\mathbf{=}\mathbf{0}$, (3)$\mathbf{+}\mathbf{2}\mathit{e}\mathit{m}\mathit{f}\mathbf{-}{\mathit{E}}_{\mathbf{2}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{3}}\mathit{L}\mathbf{=}\mathbf{0}$, (4)${\mathit{E}}_{\mathbf{1}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{2}}\mathit{L}\mathbf{=}\mathbf{0}$, (5)$\mathbf{+}\mathbf{2}\mathit{e}\mathit{m}\mathit{f}\mathbf{-}{\mathit{E}}_{\mathbf{1}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{2}}\mathit{L}\mathbf{=}\mathbf{0}$, (6)$\mathbf{+}\mathbf{2}\mathit{e}\mathit{m}\mathit{f}\mathbf{-}{\mathit{E}}_{\mathbf{1}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{3}}\mathit{L}\mathbf{=}\mathbf{0}$, (7)$\mathbf{+}\mathbf{2}\mathit{e}\mathit{m}\mathit{f}\mathbf{-}{\mathit{E}}_{\mathbf{1}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{2}}\mathit{L}\mathbf{-}{\mathit{E}}_{\mathbf{3}}\mathit{L}\mathbf{=}\mathbf{0}$. (c) It is also necessary to write charge conservation equations (node) equations. Each such equation must relate electron current flowing into a node to electron current flowing out of a node. Which of the following are valid charge conservation equations for this circuit? (1)${\mathit{i}}_{\mathbf{1}}\mathbf{=}{\mathit{i}}_{\mathbf{3}}$, (2)${\mathit{i}}_{\mathbf{1}}\mathbf{=}{\mathit{i}}_{\mathbf{2}}$, (3)${\mathit{i}}_{\mathbf{1}}\mathbf{=}{\mathit{i}}_{\mathbf{2}}\mathbf{+}{\mathit{i}}_{\mathbf{3}}$. Each battery has an emf of $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{V}$. The length of the tungsten filament in each bulb is $\mathbf{0}\mathbf{.}\mathbf{008}\mathbf{}\mathbf{m}$. The radius of the filament is$\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{m}$(it is very thin!). The electron mobility of tungsten is localid="1668588909714" $\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{}\mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}\mathbf{/}\mathbf{(}\mathbf{V}\mathbf{/}\mathbf{m}\mathbf{)}$. Tungsten has localid="1668588927161" $\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons per cubic meter. Since there are three unknown quantities, we need three equations relating these quantities. Use any two valid energy conservation equations and one valid charge conservation equation to solve for localid="1668588943223" ${\mathit{E}}_{\mathbf{1}}\mathbf{,}{\mathit{E}}_{\mathbf{2}}\mathbf{,}{\mathit{i}}_{\mathbf{1}}$and localid="1668588965567" ${\mathit{i}}_{\mathbf{2}}$.

The circuit shown in Figure 18.107 consists of a single battery, whose emf is $\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{V}$, and three wires made of the same material but having different cross-sectional areas. Each thick wire has a cross-sectional area $\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}{\mathbf{m}}^{\mathbf{2}}$and is $\mathbf{25}\mathbf{cm}$long. The thin wire has a cross-sectional area $\mathbf{5}\mathbf{.}\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}{\mathbf{m}}^{\mathbf{2}}$and is $\mathbf{6}\mathbf{.}\mathbf{1}\mathbf{cm}$long. In this metal, the electron mobility is $\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\raisebox{1ex}{$\left({\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}\right)$}\!\left/ \!\raisebox{-1ex}{$\left({\displaystyle \raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}\right)$}\right.$, and there are $\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons/m³.

(a) Which of the following statements about the circuit in the steady state are true? (1) At location B, the electric field points toward the top of the page. (2) The magnitude of the electric field at locations F and C is the same. (3) The magnitude of the electric field at locations D and F is the same. (4) The electron current at location D is the same as the electron current at location F . (b) Write a correct energy conservation (loop) equation for this circuit, following a path that starts at the negative end of the battery and goes counterclockwise. (c) Write this circuit's correct charge conservation (node) equation. (d) Use the appropriate equation(s), plus the equation relating electron current to electric field, to solve for the magnitudes E_Dand E_F of the electric field at locations D and F . (e) Use the appropriate equation(s) to calculate the electron current at location D in the steady state.

In the circuit shown figure 18.108, two thick copper wires connect a 1.5 V battery to a Nichrome wire. Each thick connecting wire is 17 cm long and has a radius of 9 mm. Copper has $\mathbf{8}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons per cubic meter and electron mobility. The Nichrome wire is 8 cm long and has a radius of 3 mm. Nichrome has $\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}$mobile electrons per cubic meter and electron mobility of $\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathbf{(}\raisebox{1ex}{${\displaystyle \raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}\mathbf{)}$}\right.}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \raisebox{1ex}{$\mathbf{(}\mathbf{V}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{m}\mathbf{)}$}\right.}$}\right.$.

(a) What is the magnitude of the electric field in the thick copper wire?

(b) What is the magnitude of the electric field in the thin Nichrome wire?