D.C. Circuits: Physics 9702 (Cambridge International AS & A Level)

A junction inside a circuit has four wires connected to it. A current of $5.0\text{ A}$ flows into the junction along one wire, and a current of $3.0\text{ A}$ flows into the junction along a second wire. A current of $6.0\text{ A}$ flows out of the junction along a third wire, and a current $I$ flows out of the junction along the fourth wire.

What is the value of $I$?

A battery of e.m.f. $12\text{ V}$ and negligible internal resistance is connected to a network of three resistors. A resistor of resistance $4.0\ \Omega$ is connected in series with a parallel combination of two resistors, of resistance $6.0\ \Omega$ and $12\ \Omega$.

(a) Calculate the combined resistance of the $6.0\ \Omega$ and $12\ \Omega$ resistors connected in parallel. [2]

(b) Calculate the total resistance of the complete network. [1]

(c) Calculate the total current supplied by the battery. [2]

(d) Calculate the current in the $6.0\ \Omega$ resistor and the current in the $12\ \Omega$ resistor, and show that these two currents are consistent with Kirchhoff's first law. [3]

A battery has e.m.f. $9.0\text{ V}$ and internal resistance $0.50\ \Omega$. It is connected to a single external resistor of resistance $4.0\ \Omega$.

(a) Write down Kirchhoff's second law for this circuit, and use it to calculate the current $I$ in the circuit. [3]

(b) Calculate the terminal potential difference of the battery. [2]

(c) Calculate the potential difference across the internal resistance (the "lost volts"). [2]

(d) Calculate the power dissipated in the internal resistance, and hence determine the efficiency of the transfer of energy from the battery to the external resistor. [3]

A potential divider circuit consists of a fixed resistor of resistance $2.0\text{ k}\Omega$ connected in series with a negative temperature coefficient (NTC) thermistor. This series combination is connected across a battery of e.m.f. $6.0\text{ V}$ and negligible internal resistance. The output potential difference $V_{\text{out}}$ is taken across the thermistor.

At room temperature, the resistance of the thermistor is $8.0\text{ k}\Omega$.

(a) State how the resistance of an NTC thermistor changes as its temperature increases. [1]

(b) Calculate $V_{\text{out}}$ at room temperature. [2]

(c) The temperature of the thermistor increases until its resistance falls to $2.0\text{ k}\Omega$. Calculate the new value of $V_{\text{out}}$. [2]

(d) State and explain how $V_{\text{out}}$ changes as the temperature of the thermistor increases, and suggest one practical device in which this potential divider circuit could be used. [3]

A single-loop circuit contains two cells and a single resistor of resistance $4.0\ \Omega$. The two cells, each of negligible internal resistance, are connected so that their e.m.f.s oppose each other: one cell has e.m.f. $6.0\text{ V}$ and the other has e.m.f. $2.0\text{ V}$, connected with reversed polarity relative to the first.

Using Kirchhoff's second law, what is the current in the circuit?

Three resistors, each of resistance $4.0\ \Omega$, are connected in parallel with each other across a battery.

What is the combined resistance of the three resistors?

A student investigates a battery of e.m.f. $E$ and internal resistance $r$ by connecting it to a single external resistor of resistance $R$ and measuring the current $I$ in the circuit.

With $R = 4.0\ \Omega$, the current is $I_1 = 2.0\text{ A}$.

The $4.0\ \Omega$ resistor is then replaced with a resistor of resistance $R = 9.0\ \Omega$, and the current becomes $I_2 = 1.0\text{ A}$.

(a) Write down the Kirchhoff's second law equation for the circuit in each of the two cases, in terms of $E$, $r$ and the given values of $R$ and $I$. [2]

(b) Solve your two equations simultaneously to find the values of $E$ and $r$. [4]

(c) Using your values from (b), calculate the terminal potential difference of the battery in the first case ($R = 4.0\ \Omega$). [2]

(d) Calculate the power dissipated inside the battery (i.e. in its internal resistance) in the second case ($R = 9.0\ \Omega$). [2]

A potential divider circuit consists of a light-dependent resistor (LDR) connected in series with a fixed resistor of resistance $1.0\text{ k}\Omega$. This series combination is connected across a battery of e.m.f. $9.0\text{ V}$ and negligible internal resistance. The output potential difference $V_{\text{out}}$ is taken across the fixed resistor.

In darkness, the resistance of the LDR is $8.0\text{ k}\Omega$.

(a) State how the resistance of an LDR changes as the light intensity falling on it increases. [1]

(b) Calculate $V_{\text{out}}$ in darkness. [2]

(c) The LDR is illuminated with bright light, causing its resistance to fall to $1.0\text{ k}\Omega$. Calculate the new value of $V_{\text{out}}$. [2]

(d) State and explain how $V_{\text{out}}$ changes as the light intensity increases, and suggest one practical device in which this potential divider circuit could be used. [3]

Two batteries, each of negligible internal resistance, are connected to a common resistor at a junction $P$, forming a two-loop network.

• A battery of e.m.f. $18\text{ V}$ is connected to $P$ through a resistor of resistance $2.0\ \Omega$, carrying current $I_1$ towards $P$.
• A battery of e.m.f. $16\text{ V}$ is connected to $P$ through a resistor of resistance $4.0\ \Omega$, carrying current $I_2$ towards $P$.
• From $P$, a single resistor of resistance $3.0\ \Omega$ carries current $I_3$ away from $P$ and back to the negative terminals of both batteries, which are joined together.

(a) State Kirchhoff's first law, and use it to write an equation relating $I_1$, $I_2$ and $I_3$ at junction $P$. [2]

(b) Use Kirchhoff's second law to write one equation for the loop containing the $18\text{ V}$ battery and the $3.0\ \Omega$ resistor, and a second equation for the loop containing the $16\text{ V}$ battery and the $3.0\ \Omega$ resistor. [2]

(c) Solve your three equations from (a) and (b) simultaneously to find $I_1$, $I_2$ and $I_3$. [5]

(d) Calculate the potential difference across the $3.0\ \Omega$ resistor. [1]

A battery of e.m.f. $15\text{ V}$ and negligible internal resistance is connected to a network of three resistors. A resistor of resistance $3.0\ \Omega$ is connected in series with a parallel combination of two resistors, of resistance $6.0\ \Omega$ and $3.0\ \Omega$.

What is the current in the $3.0\ \Omega$ resistor of the parallel combination?