Using Dimensional Analysis

Oct 13, 2025 by pinkandtrash.com 76 views

The Time Dependence of a Physical Quantity: Calculating the Dimension of α

In physics, the time dependence of a physical quantity is often described by an equation that relates the quantity to time. One such equation is the exponential function, where the quantity is given by $ P = P_0 e^{\alpha t} $. In this equation, $ P_0 $ is the initial value of the quantity, $ \alpha $ is a constant, and $ t $ is time. The dimension of $ \alpha $ is a crucial aspect of this equation, as it determines the units of the quantity. In this article, we will calculate the dimension of $ \alpha $.

The equation $ P = P_0 e^{\alpha t} $ describes the time dependence of a physical quantity. The quantity $ P $ is a function of time $ t $, and the constant $ \alpha $ determines the rate at which the quantity changes with time. To calculate the dimension of $ \alpha $, we need to understand the dimensions of the other variables in the equation.

The dimensions of the variables in the equation are as follows:

$ P $: The dimension of $ P $ is the same as the dimension of the physical quantity being described.
$ P_0 $: The dimension of $ P_0 $ is the same as the dimension of the physical quantity being described.
$ \alpha $: The dimension of $ \alpha $ is unknown and needs to be calculated.
$ t $: The dimension of $ t $ is time, which is typically denoted by $ T $.

To calculate the dimension of $ \alpha $, we can use the fact that the dimensions of the variables in the equation must be consistent. Specifically, the dimensions of the left-hand side of the equation must be the same as the dimensions of the right-hand side.

Using Dimensional Analysis

We can use dimensional analysis to calculate the dimension of $ \alpha $. The idea is to express the dimensions of the variables in terms of the fundamental dimensions of the physical quantity being described. For example, if the physical quantity is length, the fundamental dimensions are $ L $ (length) and $ T $ (time).

Applying Dimensional Analysis

Let's apply dimensional analysis to the equation $ P = P_0 e^{\alpha t} $. We can start by expressing the dimensions of the variables in terms of the fundamental dimensions.

$ P $: $ [P] = [L]^a [T]^b $
$ P_0 $: $ [P_0] = [L]^a [T]^b $
$ \alpha $: $ [\alpha] = ? $
$ t $: $ [t] = [T] $

Simplifying the Dimensions

We can simplify the dimensions of the variables by canceling out the common factors.

$ P $: $ [P] = [L]^a [T]^b $
$ P_0 $: $ [P_0] = [L]^a [T]^b $
$ \alpha $: $ [\alpha] = ? $
$ t $: $ [t] = [T] $

Equating Dimensions

We can equate the dimensions of the left-hand side and the right-hand side of the equation.

$ [P] = [P_0] e^{\alpha [t]} $
$ [L]^a [T]^b = [L]^a [T]^b e^{\alpha [T]} $

Simplifying the Exponential

We can simplify the exponential term by using the fact that $ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $.

$ [L]^a [T]^b = [L]^a [T]^b (1 + \alpha [T] + \frac{(\alpha [T])^2}{2!} + \cdots) $

Canceling Out Common Factors

We can cancel out the common factors on both sides of the equation.

$ [L]^a [T]^b = [L]^a [T]^b (1 + \alpha [T]) $

Simplifying the Equation

We can simplify the equation by canceling out the common factors.

$ [L]^a [T]^b = [L]^a [T]^b (1 + \alpha [T]) $
$ 1 = 1 + \alpha [T] $

Solving for α

We can solve for $ \alpha $ by canceling out the common factors.

$ 1 = 1 + \alpha [T] $
$ \alpha [T] = 0 $
$ \alpha = 0 $

In this article, we calculated the dimension of $ \alpha $ in the equation $ P = P_0 e^{\alpha t} $. We used dimensional analysis to express the dimensions of the variables in terms of the fundamental dimensions of the physical quantity being described. We then equated the dimensions of the left-hand side and the right-hand side of the equation and simplified the exponential term. Finally, we solved for $ \alpha $ by canceling out the common factors. The result is that $ \alpha $ has a dimension of $ T^{-1} $.

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

The dimension of $ \alpha $ is $ T^{-1} $, which means that $ \alpha $ has a unit of inverse time, such as s$^{-1} $. This result is consistent with the fact that the exponential function describes a quantity that changes with time.
Q&A: The Time Dependence of a Physical Quantity

In our previous article, we calculated the dimension of $ \alpha $ in the equation $ P = P_0 e^{\alpha t} $. In this article, we will answer some common questions related to the time dependence of a physical quantity.

Q: What is the physical significance of the dimension of α?

A: The dimension of $ \alpha $ represents the rate at which the physical quantity changes with time. In other words, it represents the rate of change of the quantity.

Q: How does the dimension of α affect the units of the physical quantity?

A: The dimension of $ \alpha $ affects the units of the physical quantity by determining the units of the rate of change. For example, if the physical quantity is length and the dimension of $ \alpha $ is $ T^{-1} $, then the units of the rate of change are $ m/s $.

Q: Can the dimension of α be negative?

A: Yes, the dimension of $ \alpha $ can be negative. This would mean that the physical quantity is decreasing with time.

Q: How does the dimension of α relate to the concept of decay?

A: The dimension of $ \alpha $ is related to the concept of decay in that it represents the rate at which a quantity decays over time. For example, if the physical quantity is the amount of a radioactive substance, then the dimension of $ \alpha $ would represent the rate at which the substance decays.

Q: Can the dimension of α be zero?

A: Yes, the dimension of $ \alpha $ can be zero. This would mean that the physical quantity is not changing with time.

Q: How does the dimension of α relate to the concept of equilibrium?

A: The dimension of $ \alpha $ is related to the concept of equilibrium in that it represents the rate at which a quantity approaches equilibrium. For example, if the physical quantity is the temperature of a system, then the dimension of $ \alpha $ would represent the rate at which the system approaches thermal equilibrium.

Q: Can the dimension of α be complex?

A: No, the dimension of $ \alpha $ cannot be complex. This is because the dimension of $ \alpha $ is a real number that represents the rate of change of the physical quantity.

Q: How does the dimension of α relate to the concept of oscillations?

A: The dimension of $ \alpha $ is related to the concept of oscillations in that it represents the rate at which a quantity oscillates over time. For example, if the physical quantity is the position of a pendulum, then the dimension of $ \alpha $ would represent the rate at which the pendulum oscillates.

In this article, we answered some common questions related to the time dependence of a physical quantity. We hope that this article has provided a better understanding of the concept of the dimension of $ \alpha $ and its relationship to the physical world.

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

The dimension of $ \alpha $ is a fundamental concept in physics that represents the rate of change of a physical quantity over time. Understanding the dimension of $ \alpha $ is essential for describing the behavior of physical systems and making predictions about their future behavior.