Question 2: Units , Dimensions and Error Analysis

Question

Consider the equation $H=\frac{x^p \epsilon^q E^r}{t^s}$
Where $H=$ magnetic field; $E=$ electric field, $\epsilon=$ permittivity, $x=$ distance, $t=$ time The values of $p, q, r$ and $s$ respectively are :

Accepted Answer

To find the values of the exponents $p, q, r$, and $s$, we will use dimensional analysis. Both sides of the equation must have the same fundamental dimensions.
The dimension of distance $(\mathrm{x})$ is the dimension of length
$$ [\mathrm{x}]=\mathrm{L} $$
The dimension is time $[\mathrm{t}]=\mathrm{T}$
The dimension of electric field ( E ) can be derived from the formula for force on a charge, $\mathrm{F}=\mathrm{qE}$. Therefore, $\mathrm{E}=\mathrm{F} / \mathrm{q}$.
The dimensions of force is $[\mathrm{F}]=\left[\mathrm{MLT}^{-2}ight]$
Charge $(\mathrm{q})=$ Current $	imes$ Time
So, the dimensions of charge is, $[\mathrm{q}]=[\mathrm{AT}]$
So, the dimensions of electric field is, $[\mathrm{E}]=\frac{\left[\mathrm{MLT}^{-2}ight]}{[\mathrm{AT}]}=\left[\mathrm{MLT}^{-3} \mathrm{~A}^{-1}ight]$
Using relationship between electric field (E) and and magnetic field (H) in electromagnetic waves. In a vacuum, the ratio of the magnitudes of the electric field to the magnetic field (B) is the speed of light (c),
$$ \mathrm{E} / \mathrm{B}=\mathrm{c} $$
Using the formula of magnetic force on a current-carrying conductor:
$$ \mathrm{F}=\mathrm{BIL} $$
Where, $\mathrm{F}=$ force $\mathrm{B}=$ magnetic field (flux density), $\mathrm{I}=$ current and $\mathrm{L}=$ length
$$ \mathrm{B}=\frac{\mathrm{F}}{\mathrm{IL}} $$
Substituting the dimensions,
$$ [\mathrm{B}]=\frac{\left[\mathrm{MLT}^{-2}ight]}{[\mathrm{LA}]} $$
$$\Rightarrow $$ $$ [\mathrm{B}]=\left[\mathrm{M} \mathrm{~T}^{-2} \mathrm{I}^{-1}ight] $$
The relation between B and H is $\mathrm{B}=\mu \mathrm{H} \Rightarrow \frac{\mathrm{B}}{\mu}=\mathrm{H}$. Where $\mu=$ permeability.
Using the formula of magnetic field, $B=\frac{\mu i}{2 \pi r} \Rightarrow\left[\frac{B}{\mu}ight]=\left[\frac{i}{r}ight]$, here $r$ is distance from straight conductor carrying current i.
$$ \left[\frac{\mathrm{B}}{\mu}ight]=[\mathrm{H}]=\frac{[\mathrm{A}]}{[\mathrm{L}]}=\left[\mathrm{L}^{-1} \mathrm{~A}ight] $$
Putting the dimensions of electric field and speed of light, the dimensions of magnetic field is,
$$ [\mathrm{H}]=\frac{\left[\mathrm{MLT}^{-3} \mathrm{~A}^{-1}ight]}{\left[\mathrm{LT}^{-1}ight]}=\left[\mathrm{MT}^{-2} \mathrm{~A}^{-1}ight] $$
We can derive the dimensions of permittivity ( $\epsilon$ ) from Coulomb's Law, $\mathrm{F}=\frac{1}{4 \pi \epsilon} \frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}^2}$.
$[\epsilon]=\left[\frac{q^2}{\mathrm{Fr}^2}ight]$
$$ [\epsilon]=\frac{[\mathrm{AT}]^2}{\left[\mathrm{MLT}^{-2}ight]\left[\mathrm{L}^2ight]}=\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2ight] $$
Substituting the dimensions into the given equation,
$$ [\mathrm{H}]=\frac{[\mathrm{x}]^{\mathrm{p}}[\epsilon]^{\mathrm{q}}[\mathrm{E}]^{\mathrm{r}}}{[\mathrm{t}]^{\mathrm{s}}} $$
$$\Rightarrow $$ $\left[L^{-1} Aight]=\frac{[L]^p\left[M^{-1} L^{-3} T^4 A^2ight]^q\left[M L T^{-3} A^{-1}ight]^r}{[T]^s}$
$$ \left[L^{-1} Aight]=\left[M^{-q+r} L^{p-3 q+r} T^{4 q-3 r-s} A^{2 q-r}ight] $$
On comparing the powers of $\mathrm{M}, \mathrm{L}, \mathrm{T}$, and A on both sides,
For $\mathrm{M},-\mathrm{q}+\mathrm{r}=0 \ldots$ (i)
For $\mathrm{A}, 2 \mathrm{q}-\mathrm{r}=1$. Since $\mathrm{r}=\mathrm{q}$, then $2 \mathrm{q}-\mathrm{q}=1$, so $\mathrm{q}=1$ and $\mathrm{r}=1$.
For $\mathrm{L}, \mathrm{p}-3 \mathrm{q}+\mathrm{r}=-1 \ldots$ (ii).
$$ \begin{gathered} p-3(1)+1=-1 \ p-2=-1 \Rightarrow p=1 \end{gathered} $$
For T, $4 q-3 r-s=0 \ldots$ (iii)
$$ \begin{gathered} 4(1)-3(1)-s=0 \ 1-s=0 \Rightarrow s=1 \end{gathered} $$
Therefore, the values are $\mathrm{p}=1, \mathrm{q}=1, \mathrm{r}=1, \mathrm{~s}=1$.
Hence, the correct option is (A).

Answer

$1, 1, 1, 1$

Answer

$-1,1,2,1$

Answer

$1, -1, -2, 1$

Answer

$-1,-2,-2,1$

Units , Dimensions and Error Analysis

Consider the equation H=\frac{x^p \epsilon^q E^r}{t^s} Where H= magnetic field; E= electric field, \epsilon= permittivity, x= distance, t= time The values of p, q, r and s respectively are :

Choose the correct answer: