Determine the dimensions of each variable.

• Specific charge $\delta$: charge per unit mass. $$[\delta] = \frac{\text{charge}}{\text{mass}} = \frac{[IT]}{[M]} = [M^{-1}IT]$$
• Potential difference $U$: energy per unit charge. Energy has dimensions of work, which is force $\times$ distance ($[MLT^{-2}] \times [L] = [ML^2T^{-2}]$). $$[U] = \frac{\text{energy}}{\text{charge}} = \frac{[ML^2T^{-2}]}{[IT]} = [ML^2T^{-3}I^{-1}]$$
• Magnetic field $B$: From the Lorentz force $F = qvB$, we have $B = \frac{F}{qv}$. $$[B] = \frac{\text{force}}{\text{charge} \times \text{velocity}} = \frac{[MLT^{-2}]}{[IT][LT^{-1}]} = \frac{[MLT^{-2}]}{[IL]} = [MT^{-2}I^{-1}]$$
• Radius $r$: length. $$[r] = [L]$$

The LHS is $\delta$.

The RHS is $\frac{2U}{B^2r^2}$. The numerical constant '2' is dimensionless.

Since $[\text{LHS}] = [M^{-1}IT]$ and $[\text{RHS}] = [M^{-1}IT]$, the dimensions on both sides are identical.
Therefore, the equation $\delta = \frac{2U}{B^2r^2}$ is homogeneous.

b)

• All correct equations must be homogeneous: This means that for any physically correct equation, the dimensions of the physical quantities on both sides of the equation must be identical. If the dimensions do not match, the equation is fundamentally incorrect and cannot describe a real physical phenomenon. Dimensional homogeneity is a necessary condition for an equation to be valid.

• Not all homogeneous equations are physically correct: An equation can be dimensionally homogeneous but still be physically incorrect. This is because dimensional analysis does not account for dimensionless numerical constants (like $\frac{1}{2}$, $2\pi$, etc.) or the precise functional relationship between variables. It only checks the consistency of units, not the accuracy of the physical law.

Example:
Consider the kinetic energy ($KE$) of an object with mass $m$ and velocity $v$.
The physically correct equation for kinetic energy is:

The dimension of kinetic energy is $[ML^2T^{-2}]$.
The dimensions of the RHS are $[M][LT^{-1}]^2 = [ML^2T^{-2}]$.
This equation is dimensionally homogeneous.

Now, consider a proposed equation for kinetic energy that is dimensionally homogeneous but physically incorrect:

The dimensions of the RHS are $[M][LT^{-1}]^2 = [ML^2T^{-2}]$.
This equation is also dimensionally homogeneous, as both sides have dimensions of $[ML^2T^{-2}]$.
However, it is physically incorrect because the numerical constant $\frac{1}{2}$ is missing. This example demonstrates that while dimensional homogeneity is a necessary condition for a correct equation, it is not a sufficient condition.