Problem 32 - Master Resource Book in JEE Main Physics

Question

The equation of alternating current is \(I=I_{0} e^{-t / C R}\) where \(t\) is time, \(C\) is capacitance and \(R\) is resistance of coil, then the dimensions of \(C R\) is (a) \(\left[\mathrm{MLT}^{-1}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{LT}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}\right]\) (d) None of these

Step-by-Step Solution

Answer

The dimensions of \( CR \) are \( [M^0 L^0 T] \), matching option (c).

Step 1: Identify Given Equation

The equation given is \( I = I_{0} e^{-t / C R} \), where \( t \) represents time, \( C \) represents capacitance, and \( R \) represents resistance.

Step 2: Analyze Exponential Expression

In an exponential function like \( e^{-t / C R} \), the exponent must be dimensionless. Hence, \(-\frac{t}{CR}\) should have no units.

Step 3: Determine Dimension of Time \( t \)

Time \( t \) has the dimension \( [T] \). This indicates \( t \) is measured in time units like seconds.

Step 4: Derive Dimensions for \( C R \)

Since \( -\frac{t}{CR} \) is dimensionless, then \( CR \) must have the same dimensions as time, \( [T] \).

Step 5: Compare Dimensions of \( C R \)

If \( CR = [T] \), then the dimensions of \( CR \) are \( [M^0 L^0 T] \). This is equivalent to option (c): \( [M^0 L^0 T] \). Therefore, these are the dimensions of \( CR \).

Key Concepts

Alternating Current

Alternating current, or AC, is a type of electrical current in which the flow of electric charge periodically reverses direction. Unlike direct current (DC) where the electric charge flows consistently in one direction, AC is more efficient for long-distance transmission because it can easily be transformed to higher or lower voltages with transformers.

Key aspects to understand about AC include:

• Frequency: This is how often the current changes direction per second. It is measured in Hertz (Hz).
• Amplitude: The peak value of the current, which indicates how much current can flow.
• Phase: Refers to the position of the current wave at a specific point in time.

The equation for alternating current given in the problem, \( I = I_0 e^{-t / CR} \), represents a decay process, which might relate to how certain circuits respond over time to an alternating current.

Capacitance

Capacitance is the ability of a system to store an electrical charge. It is a fundamental property of capacitors, which are devices used to store energy in an electric field temporarily. The capacitance is typically measured in farads (F).

• Capacitance depends on factors such as the surface area of the plates in the capacitor, the distance between them, and the properties of the dielectric material in between.
• It is mathematically expressed as: \( C = \frac{Q}{V} \), where \( Q \) is the charge, and \( V \) is the voltage.

In relation to the equation, capacitance \( C \) forms part of the term \( CR \) and affects the rate at which the alternating current decays. Understanding how capacitance influences this process is crucial for designing circuits that respond appropriately under alternating current conditions.

Resistance

Resistance is the measure of the opposition to the flow of an electric current through a conductor. It is an essential component in controlling the amount of current flowing through an electrical circuit. Resistance is measured in ohms (\(\Omega\)).

Several factors affect resistance, including:

• The material of the conductor: Some materials allow electricity to flow more freely than others.
• The size of the conductor: Longer and thinner wires tend to have higher resistance.
• Temperature: Typically, resistance increases with temperature in conductors.

In the current equation, resistance \( R \) is a key factor alongside capacitance \( C \) in determining the time characteristic \( CR \) which impacts the rate of change of the current.

Exponents in Physics

Exponents in physics are used to express extremely large or small numbers in a more manageable form and are essential in representing exponential relationships. In the context of the given equation, \( e^{-t/CR} \) is an exponential function where \( -t/CR \) is the exponent.

Why are exponents important in this context?

• They allow us to describe decay processes succinctly, which are common in alternating current circuits.
• An exponent must be dimensionless, implying that the units in the numerator and the denominator balance out.

This principle is what leads to determining the dimensions of \( CR \) as dimensionless when it equals time, aligning correctly with physical expectations and mathematical necessity for such decay equations.