Chapter 26 — Direct-Current Circuits

15 points · 3 extra credit

Chapter 26 — Direct-Current Circuits

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

1. 1. (1 pt •) Three resistors, $R_1=4\,\Omega$, $R_2=12\,\Omega$, and $R_3=6\,\Omega$, are available.

   (a) Find their equivalent resistance if all three are wired in series.

   (b) Find their equivalent resistance if all three are wired in parallel.

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2. 2. (2 pts ••) A $24\,\text{V}$ battery (assume no internal resistance) is connected to a resistor $R_1=6\,\Omega$ in series with a parallel combination of $R_2=12\,\Omega$ and $R_3=6\,\Omega$.

   (a) Find the equivalent resistance of the whole network and the total current the battery delivers.

   (b) Find the voltage across the parallel combination.

   (c) Find the current through $R_2$ and through $R_3$.

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3. 3. (2 pts ••) A battery of emf $\mathcal{E}=12\,\text{V}$ has internal resistance $r=0.5\,\Omega$ and is connected to an external resistor $R=5.5\,\Omega$.

   (a) Find the current in the circuit.

   (b) Find the terminal voltage of the battery.

   (c) Find the power dissipated in $R$ and the power dissipated inside the battery.

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4. 4. (2 pts ••) An ammeter and a voltmeter are idealized measuring instruments.

   (a) An ideal ammeter measures the current through a branch. Should it have very large or very small resistance, and should it be connected in series or in parallel with that branch? Explain what would go wrong if its resistance were not ideal.

   (b) An ideal voltmeter measures the potential difference across an element. Should it have very large or very small resistance, and should it be connected in series or in parallel with that element? Explain what would go wrong if its resistance were not ideal.

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5. 5. (2 pts ••) A capacitor $C=50\,\mu\text{F}$, initially uncharged, is connected in series with a resistor $R=100\,\text{k}\Omega$ and a $\mathcal{E}=9\,\text{V}$ battery through a switch that closes at $t=0$.

   (a) Find the time constant $\tau$ and the initial charging current $i(0)$.

   (b) Find the final charge on the capacitor once it is fully charged.

   (c) Find the charge on the capacitor and the current in the circuit at $t=10\,\text{s}$.

   (d) How long after closing the switch does the charge reach $90\%$ of its final value?

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6. 6. (2 pts ••) A capacitor $C=100\,\mu\text{F}$ is charged to $V_0=12\,\text{V}$ and then, at $t=0$, connected across a resistor $R=20\,\text{k}\Omega$ (the battery is removed) so it discharges.

   (a) Find the time constant and the initial discharge current.

   (b) Find the voltage across the capacitor at $t=3\,\text{s}$.

   (c) Find the energy stored in the capacitor at $t=0$. Where does this energy go as the capacitor discharges?

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7. 7. (1 pt •) In the charging circuit of Problem 5, at the instant the switch closes ($t=0$) the capacitor is uncharged.

   (a) What is the potential difference across the capacitor at that instant, and what is the potential difference across the resistor?

   (b) After a very long time, what is the current in the circuit, and what fraction of the battery's total energy output has been stored in the capacitor versus dissipated in the resistor?

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8. 8. (3 pts •••) Two batteries and three resistors form the following two-loop network. The left branch contains a battery $\mathcal{E}_1=12\,\text{V}$ in series with $R_1=4\,\Omega$; the middle branch contains a battery $\mathcal{E}_2=9\,\text{V}$ in series with $R_2=3\,\Omega$; the right branch contains only $R_3=6\,\Omega$. All three branches run between the same top node $a$ and bottom node $b$, with each battery's $+$ terminal toward node $a$.

   (a) Define a current in each branch and write Kirchhoff's junction rule at node $a$.

   (b) Write two independent loop equations and solve for all three branch currents. State the direction of each current (toward or away from node $a$).

   (c) For each battery, determine whether it is delivering energy or absorbing energy (being charged), based on whether its current leaves or enters its $+$ terminal. Then verify overall energy balance: show that the total power delivered by the sources equals the total power dissipated in the three resistors.

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9. Extra Credit (3 pts •••)

   Derive the charging law for the series R–C circuit of Problem 5 from first principles.

   (a) Apply Kirchhoff's loop rule at an arbitrary time $t$ to write the first-order differential equation governing the charge $q(t)$ on the capacitor, using $i=dq/dt$.

   (b) Solve the differential equation by separation of variables with the initial condition $q(0)=0$, and show that

   $$q(t)=C\mathcal{E}\left(1-e^{-t/RC}\right).$$

   (c) Differentiate to obtain $i(t)$, and prove by direct integration that the total energy delivered by the battery over all time, $\int_0^{\infty}\mathcal{E}\,i\,dt$, splits into exactly $\tfrac12 C\mathcal{E}^2$ stored in the capacitor and $\tfrac12 C\mathcal{E}^2$ dissipated in the resistor — independent of the value of $R$.

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