The dimensions of ${\left({\mu }_0{\epsilon }_0\right)}^{-1/2}$ are:

1.  $[$ $L^{-1}T$ $]$

2.  $[$ $L{T}^{-1}$ $]$

3.  $[$ $L^{1/2}{T}^{1/2}$ $]$

4. $[$ $L^{1/2}{T}^{-1/2}$ $]$

The density of a material in a CGS system of units is \(4~\text{grams/cm}^3\). In a system of units in which the unit of length is \(10~\text{cm}\) and the unit of mass is \(100~\text{grams}\), the value of the density of the material will be:
1. \( 0.04 \)
2. \( 0.4 \)
3. \( 40 \)
4. \(400\)

A student measures the distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimate \(g,\) the acceleration due to gravity. If the maximum percentage errors in the measurement of the distance and the time are \(e_1\) and \(e_2\) respectively, the percentage error in the estimation of \(g\) is:
1. $e_1+2e_2$
2. $e_1+e_2$
3. $e_1-2e_2$
4. $e_2-e_1$

The velocity \(v\) of a particle at time \(t\) is given by \(v=at+\dfrac{b}{t+c}\), where \(a,\) \(b\) and \(c\) are constants. The dimensions of \(a,\) \(b\) and \(c\) are respectively:
1. \(\left[{LT}^{-2}\right],[{L}] \text { and }[{T}]\)
2. \( {\left[{L}^2\right],[{T}] \text { and }\left[{LT}^2\right]}  \)
3. \( {\left[{LT}^2\right],[{LT}] \text { and }[{L}]}  \)
4. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)

Dimensions of resistance in an electrical circuit, in terms of dimension of mass M, length L, time T, and current I, would be:

1. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{I}}^{-1}\right]$

2. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]$

3. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}{\mathrm{I}}^{-1}\right]$

4. $\left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{I}}^{-2}\right]$

If the error in the measurement of the radius of a sphere is 2%, then the error in the determination of the volume of the sphere will be:

1. 4%

2. 6%

3. 8%

4. 2%

The dimensions of $\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{E}}^2$ where ${\epsilon }_0$ is the permittivity of free space and E is the electric field, are:

1. [ML²T^-2]

2. [ML^-1T^-2]

3.  [ML²T^-1]

4.  [MLT^-1]