In terms of potential difference $V$,electric current $I$,permittivity $\varepsilon_0$,permeability $\mu_0$,and speed of light $c$,the dimensionally correct equation$(s)$ is(are):
$(A)$ $\mu_0 I^2 = \varepsilon_0 V^2$
$(B)$ $\varepsilon_0 I = \mu_0 V$
$(C)$ $I = \varepsilon_0 cV$
$(D)$ $\mu_0 cI = V$

Heat produced in a current-carrying conducting wire depends on current $I$,resistance $R$ of the wire,and time $t$ for which the current is passed. Using these facts,obtain the formula for heat energy.

If $\varepsilon_0$ is the permittivity of free space,$e$ is the charge of a proton,$G$ is the universal gravitational constant,and $m_p$ is the mass of a proton,then the dimensional formula for $\frac{e^2}{4 \pi \varepsilon_0 G m_p^2}$ is:

From the equation $\tan \theta = \frac{rg}{v^2}$,one can obtain the angle of banking $\theta$ for a cyclist taking a curve (the symbols have their usual meanings). Then say,it is

The formula is $X = 5YZ^2$,where $X$ and $Z$ have dimensions of capacitance and magnetic field,respectively. What are the dimensions of $Y$ in $SI$ units?

$A$ wave pulse can travel along a tense string like a violin string. $A$ series of experiments showed that the wave velocity $V$ of a pulse depends on the following quantities: the tension $T$ of the string,the cross-sectional area $A$ of the string,and the density $\rho$ (mass per unit volume) of the string. Obtain an expression for $V$ in terms of $T$,$A$,and $\rho$ using dimensional analysis.