Question

Two resistors, \(R_{1}\) and \(R_{2},\) are connected in series across a potential difference, \(\Delta V_{0}\). Express the potential drop across each resistor individually, in terms of these quantities. What is the significance of this arrangement?

Short answer

Answer: The potential difference in a series arrangement of two resistors is distributed between the resistors such that ΔV1 = (ΔV0 / (R1 + R2)) * R1 and ΔV2 = (ΔV0 / (R1 + R2)) * R2. The significance of this arrangement is that it allows for the control of overall resistance and the current passing through the circuit. Additionally, it helps in distributing power among various components and ensures the conservation of energy within the circuit, as the sum of individual voltage drops equals the applied voltage.

Step-by-step solution

Identify the resistors connected in series

In this circuit, two resistors, \(R_{1}\) and \(R_{2}\), are connected in series. It means they are connected end-to-end, with only one path for electric current to flow through them. This type of connection has the property where the current passing through the resistors will be the same.

Calculate the total resistance

For resistors connected in series, the total resistance, \(R_{total}\), can be found by adding the resistances of all the resistors. Mathematically, the total resistance can be expressed as: \(R_{total} = R_{1} + R_{2}\)

Calculate the current flowing through the circuit

Next, we need to find the total current, \(I\), that is flowing through the circuit. We can do this by applying Ohm's law, which states that the current flowing through a circuit is directly proportional to the potential difference across the circuit and inversely proportional to the resistance of the circuit: \(I = \frac{\Delta V_{0}}{R_{total}}\) Using the expression for \(R_{total}\) from Step 2, we can rewrite the formula as: \(I = \frac{\Delta V_{0}}{R_{1} + R_{2}}\)

Calculate the potential drop across each resistor

Now that we have calculated the current, we can use Ohm's law again to find the potential drop, \(\Delta V_{1}\) and \(\Delta V_{2}\), across each resistor individually: \(\Delta V_{1} = I * R_{1}\) \(\Delta V_{2} = I * R_{2}\) Substituting the value of the current \(I\) from step 3 into these expressions, we get: \(\Delta V_{1} = (\frac{\Delta V_{0}}{R_{1} + R_{2}}) * R_{1}\) \(\Delta V_{2} = (\frac{\Delta V_{0}}{R_{1} + R_{2}}) * R_{2}\)

Explain the significance of the arrangement

The obtained expressions for the individual potential drops, \(\Delta V_{1}\) and \(\Delta V_{2}\), indicate how the total potential difference \(\Delta V_{0}\) is distributed between the resistors in the series connection. Depending on the magnitude of the resistances, each resistor will have a different potential drop, and thus, will dissipate different amounts of power. This can be useful for designing circuits where power needs to be distributed among various components, ensuring proper function of the devices. Additionally, this arrangement allows us to control the overall resistance and consequently the current passing through the circuit. There is conservation of energy within this arrangement, where the sum of the individual voltage drops will be equal to the applied voltage. It showcases the trade-off between voltage and resistance keeping the current constant in a series connection.

Key concepts

Ohm's Law

At the heart of understanding electric circuits is Ohm's Law, a fundamental concept that relates electric current, resistance, and potential difference (voltage). It can be simply stated as:

\( I = \frac{\Delta V}{R} \)

where \( I \) represents the current in amperes, \( \Delta V \) is the potential difference in volts, and \( R \) is the resistance in ohms. Ohm's Law explains that the electric current flowing through a conductor between two points is directly proportional to the potential difference across the two points and inversely proportional to the resistance.

For example, if you increase the potential difference across a conductor, the current will also increase, provided the resistance remains constant. Conversely, increasing the resistance will result in a decrease in the current. This law is crucial when analyzing circuits, as it allows us to determine any of these three values if the other two are known.

Series Circuit Analysis

In a series circuit, components are connected end-to-end, forming a single path for the electric current to flow. The most significant attribute of series circuits is that the same current flows through all components.

When performing a series circuit analysis, we aim to understand the behavior of current and how the potential difference (voltage) is distributed across each component. The total resistance of a series circuit is the sum of the individual resistances, which directly affects the current according to Ohm's Law. This can be expressed as:

\( R_{total} = R_{1} + R_{2} + \ldots + R_{n} \)

Knowing the total resistance enables us to calculate the total current and the voltage drop across each resistor using Ohm's Law. Since the current is constant throughout the series circuit, the voltage drop across each resistor will depend on its resistance. This principle is vital for understanding why different devices in a series circuit will have different potential differences across them.

Potential Difference

Potential difference, commonly referred to as voltage, is the driving force that compels electric charge to move through a circuit, thus generating an electric current. The potential difference across a component in a circuit is the amount of energy provided or taken per unit charge as it passes through the component.

In the context of series circuits, the total potential difference of the power source is divided among the series components. Each resistor's potential difference is calculated using Ohm's Law, shown as \( \Delta V = I \times R \), where \( I \) is the current and \( R \) is the resistance. The sum of the voltage drops equals the total potential difference provided by the power source, aligning with the law of conservation of energy. This concept is fundamental to understanding how power is distributed in a circuit and how to manipulate these distributions for desired outcomes.

Electric Current

Electric current is the rate at which charge flows through a point in a circuit and is measured in amperes (A). In the context of resistors in series, it's crucial to recognize that the current is the same at every point in the series circuit. This means that no matter how many resistors you have in series or their values, the current that flows must pass through each resistor without any loss.

The uniformity of current in a series circuit makes troubleshooting simpler. For instance, if a part of the circuit breaks or a component fails, the entire circuit stops functioning, indicating a problem. The concept of electric current is essential when analyzing circuits because it influences how we calculate other important quantities like power dissipation and potential difference across components.