Chapter 25 — Current, Resistance & Electromotive Force

15 points · 3 extra credit

Chapter 25 — Current, Resistance & Electromotive Force

Core: 15 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (2 pts ••) A copper wire of diameter $2.05\,\text{mm}$ carries a steady current of $8.0\,\text{A}$. Copper has $n=8.5\times10^{28}$ free electrons per cubic metre.

   (a) Find the current density $J$ in the wire.

   (b) Find the drift speed of the conduction electrons.

   (c) How many electrons pass through a cross-section of the wire in one minute?

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2. 2. (2 pts ••) A heating element is made from nichrome wire ($\rho=1.10\times10^{-6}\,\Omega\!\cdot\!\text{m}$) of length $3.5\,\text{m}$ and diameter $0.50\,\text{mm}$.

   (a) Find the resistance of the element at room temperature.

   (b) A second element uses the same material but with twice the length and half the diameter. By what factor does its resistance change relative to part (a)? Give the new resistance.

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3. 3. (2 pts ••) A copper coil has a resistance of $25.0\,\Omega$ at $20\,{}^\circ\text{C}$. The temperature coefficient of resistivity for copper is $\alpha=3.9\times10^{-3}\,\text{K}^{-1}$, and over this range you may treat resistance as varying linearly with temperature.

   (a) Find the resistance of the coil when it heats up to $85\,{}^\circ\text{C}$.

   (b) At what temperature would the coil's resistance read $30.0\,\Omega$?

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4. 4. (1 pt •) A resistor is connected across an ideal $12\,\text{V}$ source and obeys Ohm's law with $R=470\,\Omega$.

   (a) Find the current through the resistor.

   (b) Find the power it dissipates.

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5. 5. (2 pts ••) A single AA cell has an emf of $1.50\,\text{V}$ and internal resistance $r=0.35\,\Omega$. It is connected to an external resistor $R=4.5\,\Omega$.

   (a) Find the current in the circuit.

   (b) Find the terminal voltage of the cell.

   (c) Find the power dissipated in the internal resistance and the power delivered to $R$, and confirm they sum to the total power the emf supplies.

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6. 6. (2 pts ••) A household space heater is rated $1500\,\text{W}$ when operated at $120\,\text{V}$.

   (a) Find the operating resistance of the heater and the current it draws.

   (b) If the heater runs $2.5$ hours per day for $30$ days and electricity costs \0.14$ per kilowatt-hour, find the total energy used (in kWh) and the cost.

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7. 7. (2 pts ••) In a cylindrical conductor of radius $R=1.2\,\text{mm}$, the current is not uniform: the current density points along the axis and has magnitude $J(r)=J_0\,(r/R)$, where $r$ is the distance from the axis and $J_0=1.5\times10^{6}\,\text{A/m}^2$.

   (a) By integrating over the cross-section, find the total current carried by the conductor. (Hint: use a thin annular ring of radius $r$ and thickness $dr$.)

   (b) Find the average current density $I/(\pi R^2)$ and compare it to $J_0$.

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8. 8. (2 pts ••) A battery of emf $\mathcal{E}$ and internal resistance $r=0.50\,\Omega$ drives an external resistor $R$. When $R=3.0\,\Omega$ the current is $2.4\,\text{A}$.

   (a) Find the emf $\mathcal{E}$.

   (b) The external resistor is then removed and the battery's terminals are short-circuited (external resistance $\to 0$). Find the short-circuit current.

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9. Extra Credit (3 pts •••)

   A solid conductor is shaped as a truncated cone (frustum) of length $L$. Its circular cross-section has radius $a$ at the left face and grows linearly to radius $b>a$ at the right face. The material has uniform resistivity $\rho$, and current flows along the axis. Assume the current density is uniform across each cross-section (a good approximation when the taper is gentle), so each thin disc acts as a resistor in series.

   (a) Let $x$ measure distance from the left face. Write the radius $\varrho(x)$ of the cross-section as a function of $x$, and hence the resistance $dR$ of a thin disc of thickness $dx$.

   (b) Integrate to show that the total resistance is

   $$R=\frac{\rho L}{\pi a b}.$$

   (c) Evaluate $R$ for a copper frustum ($\rho=1.7\times10^{-8}\,\Omega\!\cdot\!\text{m}$) with $L=0.15\,\text{m}$, $a=1.0\,\text{mm}$, and $b=3.0\,\text{mm}$. Check that your formula reduces to $\rho L/(\pi a^2)$ when $b\to a$ (a uniform cylinder).

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