Validity of dimensional analysis in theoretical physics My textbook mentions the following lines about the validity of dimensional analysis.

..... if an equation fails this consistency test, it is proved wrong but if it passes it is not proved right. Thus a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong equation or inconsistent equation must be wrong

As far as i understood there lines, it meant that if an equation is true dimensionally then it need not be true in general. So dimensional analysis doesn't prove equations in physics but is a powerful tool while analysing dimensions.

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*So are there any examples to this?


*Are there any examples where some equation is dimensionally true but isn't true in general?
 A: A simple example: $x=at^2$ is dimensionally homogeneous, but the true equation is $x=\frac{1}{2}at^{2}\;\;$ (in the case $\;v_{0}=0,x_{0}=0$).
A: An especially awkward situation occurs when something is dimensionless. In special relativity, $\gamma$ is a function of $\beta:=v/c$, but dimensional analysis doesn't constrain $\gamma$ at all, except to ban dimensionful coefficients. For example, we can't have $\gamma=m\beta$ with $m$ a mass, but for all we know$$\gamma=\frac{\arcsin^3\beta\cot\zeta^5(-\ln\beta)}{1+\exp\sqrt{19+\beta^2}}.$$You actually need some, you know, theory to realize instead$$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$is the only possibility.
A: Yes, dimensional analysis is primarily used for verification, and for the interconversion of units, but it cannot be used to properly derive an equation.
Take the simple example of the area '$A$'of a circle with radius '$r$'. If you solely use dimensional analysis to establish a relationship between the two, you will find that:
$$[A] = L^2 \text{ } ; \text{ }[r] = L \rightarrow A = kr^2, k \in \mathbb{R}$$
Now from the derived equation, there is no way to fix the value of the constant '$k$' with dimensions alone, so you can't say if the area is $A = 50r^2$ or $A = \pi r^2$.
However, if a source claims that the area is $A = 4r^3$, it can be clearly shown that the said source is incorrect dimensionally.
Note that $A = 5r^2$, $A = 3.14r^2$ and $A = \pi r^2$ are all dimensionally correct, but only the last one is true in general, for a circle.
In the Bohr's model of the Hydrogen Atom, one of the postulates had no basis in classical physics: $$m_evr = n\hbar $$
At the time, this could be verified dimensionally because angular momentum ($L = m_evr$) and the reduced plank's constant ($\hbar$) had the same dimensions. This equation however, is incorrect, despite being dimensionally correct.
Hope this helps.
A: You can actually do much more than just checking answers with dimensional analysis.
Consider a simple pendulum in a gravitational field. There are only three parameters in this system: the mass of the pendulum (m), the length of the pendulum (L), and the acceleration due to gravity (g).
(We are assuming that friction is irrelevant, and that the gravitational field doesn’t change perceptibly from across the pendulum. These are exactly the assumptions made in intro physics textbook solutions to this problem.)
Now, we can ask, what is the period of the pendulum? We can set up the differential equation form of F=ma and solve this, of course. But we can actually do better than this. What quantity with units of time can I make out m, L, and g? The only one is √(L/g).
In fact the right answer is 2π √(L/g) , so we are off by a factor “of order one” as is typical of these situations. But we’ve still learned a lot, in terms of how the variables must actually enter. Perhaps more importantly, the expression tells us one very important bit of physics: the mass doesn’t matter.
This same sort of analysis works in other contexts and can lead to all sorts of insights.
