Animated Solution for Physics - Physics and Measurement: The dimension of stopping potential $V_0$ in photoelectric effect in units of Planck's constant $h$, speed of light $c$ and gravitational constant $G$ and ampere $A$ is

Imagine you are given a completely new physical quantity and asked to find its relationship with other fundamental constants. It sounds like a daunting task, right? But physics gives us a superpower: Dimensional Analysis.

In this problem, we are asked to express the stopping potential $V_0$ in terms of Planck's constant $h$, the speed of light $c$, the gravitational constant $G$, and electric current $A$.

Let's dive into this beautiful algebraic puzzle!

Before we build our master equation, we need to know the building blocks. We must find the dimensional formula for each quantity involved.

1. Stopping Potential ($V_0$): Potential is defined as the work done per unit charge.

2. Planck's Constant ($h$): From the famous equation $E = h u$, we know $h$ is energy divided by frequency.

3. Speed of Light ($c$): This is simply velocity.

4. Gravitational Constant ($G$): Using Newton's law of gravitation $F=\frac{G{m}_1{m}_2}{r^2}$, we can isolate $G$.

Now, we assume that the stopping potential $V_0$ is proportional to some unknown powers of $h$, $c$, $G$, and $A$. Let's call these powers $a$, $b$, $c$, and $d$.

By the principle of dimensional homogeneity, the dimensions on the left-hand side must perfectly match the dimensions on the right-hand side. Let's substitute our derived dimensional formulas into this equation:

Now, we group the bases $M$, $L$, $T$, and $A$ on the right side by adding their exponents:

To maintain balance, we equate the exponents of corresponding base quantities from both sides. This gives us a system of four linear equations:

For $M$: $a-c=1$ For $L$: $2a+b+3c=2$ For $T$: $-a-b-2c=-3$ For $A$: $d=-1$

We already have our first victory: $d=-1$.

Next, let's look at the equations for $L$ and $T$. If we add them together, the variable $b$ magically cancels out!

Now we have a simple system of two equations with two variables: $a-c=1$ $a+c=-1$

Adding these two equations yields $2a=0$, which means $a=0$. Substituting $a=0$ back into $a-c=1$ gives us $c=-1$.

Finally, we substitute $a=0$ and $c=-1$ into the equation for $L$:

We have successfully conquered the algebra! Our exponents are $a=0$, $b=5$, $c=-1$, and $d=-1$.

Substituting these back into our original assumed relation, we get the final dimensional formula:

If you look closely at the options provided in the original exam question, you will notice that none of them match our rigorously derived result.

This happens sometimes in competitive exams! The key takeaway is to always trust your mathematical process. When you follow the laws of physics and algebra flawlessly, you can be confident in your answer, even when the options try to deceive you.