Question

The current through each resistor in the two-resistor circuit is _________ the current through the resistor in the one-resistor circuit (the circuit in Part A). The voltage across each resistor in the two-resistor circuit is ___________ the voltage across the resistor in the one-resistor circuit.

• the same as / half
• half / half
• twice / half
• half / twice
• half / the same as
• twice / twice

Answer 1

Concepts and reason The concept required to solve the question is Ohm's law, series, and parallel combinations of resistors. According to the resistors' arrangements, apply Ohm's law to find the current through resistors. Then again, apply Ohm's law to find the voltage across the resistor.

Fundamentals

Ohm's law states that the potential difference across a circuit is proportional to the current and the resistance applied across it.

It is mathematically given by, \(V=I R\)

Here, \(\mathrm{V}\) is voltage, \(\mathrm{I}\) is current, and \(\mathrm{R}\) is resistance.

The current through the circuit in which only one resistance is connected is given by Ohm's law. It is mathematically given by, \(V=I R\)

Rearrange the equation to find current. \(I_{1}=\frac{V}{R} \ldots \ldots\) (1)

Here, \(\mathrm{V}\) is the voltage, 11 is current when one resistance is connected in the circuit, and \(\mathrm{R}\) is the resistance. Now, the current through the circuit in which two resistances are connected is given by Ohm's law. It is mathematically given by,

$$ \begin{aligned} I_{2} &=\frac{V}{R+R} \\ &=\frac{V}{2 R} \end{aligned} $$

Here, the value of resistance will be twice because two resistances are connected in series, \(V 2\) is the voltage, 12 is the current when two resistances are connected in the circuit. So, the value of current is \(I_{2}=\frac{V}{2 R} \ldots \ldots(2)\)

Divide the currents of equations (1) and (2).

$$ \frac{I 1}{I 2}=\frac{(V / R)}{(V / 2 R)} $$

\(\frac{I_{1}}{2}=I_{2}\)

Thus, the current resistor in the two resistor circuit is half the current through the resistor in the one resistor circuit.

According to Ohm's law, the current through the circuit is directly proportional to the voltage across the circuit and inversely proportional to the resistance connected in the circuit.

According to Ohm's law, voltage and current are directly proportional. The equation of Ohm's law is, \(V=I R\)

So, according to the first step, where the current will become half if two resistors are connected, the voltage will also become half. Thus, the voltage through each resistor in the two resistors circuit is half the voltage through the resistor in the one resistor circuit.

According to the Ohm's law, voltage and current are directly proportional. So, if the current through each resistor in the two resistors circuit is half, then the voltage will also become half.

The current and the voltage in the two-resistor system are half and a half, respectively.