Ohm's Law

1. Find the current through the ammeter.
2. Find the energy output by the 12 V battery in 3 s.
3. Find the total heat generated in this time. Why are the results of (2) and (3) different?

1. Find the current through $\varepsilon_3$.
2. Find the power dissipated in $R_2$.
3. Find the power delivered by $\varepsilon_3$ to the external circuit.
4. Find the total rate of heat generation in the circuit.

1. Given $\varepsilon = 6$ V, $R_1 = 100$ $\Omega$, $R_2 = 200$ $\Omega$, $R_3 = 300$ $\Omega$, $R_4 = 400$ $\Omega$, and $R_g = 500$ $\Omega$, find $I_1, I_2, I_3, I_4$, and $I_g$.
2. Prove that when the arm resistances satisfy $R_1/R_2 = R_3/R_4$, the galvanometer current $I_g = 0$. (This is the balanced-bridge condition.)

Two resistance wires of the same material are connected in two separate circuits. Wire A has length $l$ and diameter $d$; wire B has length $3l$ and diameter $3d$.

To make both wires produce equal heat in the same time, what should be the ratio of voltages applied to them, $U_A : U_B$?

A. $U_A : U_B = 1 : 1$

B. $U_A : U_B = \sqrt{3} : 3$

C. $U_A : U_B = \sqrt{3} : 1$

D. $U_A : U_B = 3 : 1$

Three bulbs $A$, $B$, $C$ are all lit in a circuit. Bulb $A$ is in series with the battery and the rest of the circuit. After $A$, the circuit splits into two parallel branches: one branch contains bulb $B$ in series with the active portion of a sliding rheostat (the portion of the rheostat between the slider $P$ and its lower terminal is used), and the other branch contains bulb $C$ alone.

When the slider $P$ of the rheostat is moved downward, how do the brightnesses of the three bulbs change?

A. $A$, $B$, $C$ all get brighter

B. $A$, $B$ get brighter; $C$ gets dimmer

C. $A$, $C$ get brighter; $B$ gets dimmer

D. $A$ gets brighter; $B$, $C$ get dimmer

Two identical batteries are connected in series with a load resistor $R$, forming a single loop. Each battery has EMF $\varepsilon = 2 \text{ V}$ and internal resistance $r = 0.1 \text{ Ω}$. The load resistance is $R = 4.8 \text{ Ω}$. Point $a$ is at the negative terminal of the first battery, and point $b$ is at the junction between the two batteries (the positive terminal of the first battery).

What is the potential difference $U_a - U_b$ between points $a$ and $b$?

A. $1.92 \text{ V}$

B. $2.0 \text{ V}$

C. $-2.0 \text{ V}$

D. $-1.92 \text{ V}$

1. What is the current when it operates normally?
2. What is its resistance?
3. If the effect of temperature on the resistance is neglected, what is its power when connected to a $110$ V supply?

Two bulbs $A$ and $B$ are both rated at $110 \text{ V}$, with rated powers $P_A = 100 \text{ W}$ and $P_B = 40 \text{ W}$. They are to be connected to a $220 \text{ V}$ supply using a variable resistor (rheostat), so that both bulbs operate at their rated values. Four possible connection schemes are shown:

Scheme A: Rheostat in parallel with the series combination ($A$ in series with $B$).

Scheme B: Bulb $B$ in series with the parallel combination (rheostat in parallel with bulb $A$).

Scheme C: Bulb $A$ in series with the parallel combination (rheostat in parallel with bulb $B$).

Scheme D: Rheostat in series with the parallel combination (bulb $A$ in parallel with bulb $B$).

Which scheme makes both bulbs operate normally while minimizing the total power consumed by the circuit?

A. Scheme A

B. Scheme B

C. Scheme C

D. Scheme D

In the circuit, every resistor has the same resistance $r$ and the same rated power. Two parallel branches are connected across the source voltage $U$:

Branch 1: $R_1$ in parallel with $R_2$ (between the left terminal and node $M$), then $R_5$ from $M$ to the right terminal.

Branch 2: $R_3$ in series with $R_4$, between the left and right terminals.

When the source voltage $U$ is gradually increased, which resistor will burn out first?

A. $R_1$ and $R_2$

B. $R_3$

C. $R_4$

D. $R_5$

A battery with EMF $\varepsilon$ and internal resistance $r$ drives two parallel branches:

Branch 1: resistor $R_1$ in series with an ammeter (treated as ideal, with negligible resistance).

Branch 2: resistor $R_2$ in series with switch $S$.

Initially, switch $S$ is closed.

When switch $S$ is opened, how does the ammeter reading change?

A. If $r = 0$, the reading increases; if $r e 0$, the reading decreases.

B. If $r = 0$, the reading decreases; if $r e 0$, the reading increases.

C. The reading increases in both cases.

D. If $r = 0$, the reading is unchanged; if $r e 0$, the reading increases.