Question

An alternating current of \(\mathrm{rms}\) value \(10 \mathrm{~A}\) is passed through a \(12 \Omega\) resistance. The maximum potential difference across the resistor is, (a) \(20 \mathrm{~V}\) (b) \(90 \mathrm{~V}\) (c) \(169.68 \mathrm{~V}\) (d) None of these

Short answer

The maximum potential difference across the resistor is \(169.68 \mathrm{~V}\). To find this, we used the relationship between rms current and peak current and then applied Ohm's Law.

Step-by-step solution

Identify the given values

We are given the following values: - rms current (I_rms) = 10 A (ampere) - Resistance (R) = 12 Ω (ohm)

Find the peak current (I_peak)

To find the peak current, we need to use the relationship between rms current and peak current. The formula to find the peak current is: \(I_{peak} = \sqrt{2} \times I_{rms}\) Substitute the given rms current value: \(I_{peak} = \sqrt{2} \times 10\) Now, calculate the peak current: \(I_{peak} = 14.14 \mathrm{~A}\)

Calculate the maximum potential difference (V_max)

Now we have the peak current (I_peak), and we can use Ohm's Law to calculate the maximum potential difference (V_max) across the resistor. Ohm's Law states that: \(V = I \times R\) Here, V represents the potential difference, I represents the current, and R represents the resistance. Since we want to find the maximum potential difference, we need to use the peak current value instead of the rms current value: \(V_{max} = I_{peak} \times R\) Substitute the values of I_peak and R: \(V_{max} = 14.14 \times 12\) Calculate the maximum potential difference: \(V_{max} = 169.68 \mathrm{~V}\)

Identify the correct answer

Comparing the calculated V_max value with the given options, we can see that the maximum potential difference across the resistor is: (c) 169.68 V

Key concepts

RMS Current

Root Mean Square (RMS) current is a way to express alternating current (AC) values in a form that can be compared directly with direct current (DC) values. This is because AC values constantly change over time, swinging from positive to negative. RMS current provides a mean or average value that represents the effective DC current that would deliver the same power to a resistor.

To calculate RMS current, you take the square of the current values over one complete cycle of the AC waveform, find the average of these square values, and finally take the square root of this average. In mathematical terms:

• RMS Value, \( I_{rms} = rac{I_{peak}}{ ext{√2}} \) for sinusoidal waveforms.

Understanding RMS is crucial because most AC instruments like wattmeters and ammeters are calibrated to read RMS values, allowing for easy power consumption calculation.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that relates voltage (V), current (I), and resistance (R) in electrical circuits. It states that the voltage across a conductor is directly proportional to the current flowing through it, provided the temperature remains constant.

The mathematical form of Ohm's Law is:

• \( V = I \times R \)

where:

• \( V \) is the voltage across the conductor.
• \( I \) is the current flowing through the conductor.
• \( R \) is the resistance of the conductor.

This law is fundamental to solving many problems, like determining the maximum voltage across components in AC circuits when given values like peak current.

Peak Current

The peak current is the maximum instantaneous value of a current waveform. Unlike RMS current, the peak current represents the highest point on the current waveform. For sinusoidal AC circuits, the peak current is particularly important in calculating factors like maximum voltage across components, as components must be able to withstand these maximum values to avoid damage.

To find the peak current, we use the relationship between RMS and peak values:

• \( I_{peak} = \sqrt{2} \times I_{rms} \)

The factor \( \sqrt{2} \) comes from the nature of sinusoidal waveforms, where it is the ratio between peak and RMS values.

Resistance

Resistance is a measure of how much a component resists the flow of electric current. It is measured in ohms (Ω) and is a crucial parameter in electrical circuits because it helps determine how current and voltage interact according to Ohm's Law. High resistance means the material or component allows less current to pass through for a given voltage, while low resistance means more current can flow.

Key points about resistance include:

• It depends on material, length, and cross-sectional area.
• Resistors are commonly used in circuits to control current flow.
• Resistance affects power dissipation as heat, calculated by \( P = I^2 \times R \).

In the context of AC circuits, knowing the resistance helps determine the power usage and the potential differences across circuit components given certain current levels.