Question

The magnetic field (dB) due to smaller element (dl) at a distance \[\overset{\to }{\mathop{r}}\,\] from element carrying current i, is
\[\begin{align}
  & A.\,dB=\dfrac{{{\mu }_{0}}i}{4\pi }\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{r} \right) \\
 & B.\,dB=\dfrac{{{\mu }_{0}}i}{4\pi }{{i}^{2}}\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{{{r}^{2}}} \right) \\
 & C.\,dB=\dfrac{{{\mu }_{0}}i}{4\pi }{{i}^{3}}\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{2{{r}^{2}}} \right) \\
 & D.\,dB=\dfrac{{{\mu }_{0}}}{4\pi }i\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{{{r}^{3}}} \right) \\
\end{align}\]

Answer 1

Hint: The question is based on the concept of Biot-Savart’s law. This law gives the equation of the magnetic field produced due to the current-carrying element. This law is applied for the symmetrical current distribution and for the conductors of small size carrying current.

Complete step by step answer:
From the given information, we have the data as follows.
The magnetic field (dB) due to smaller element (dl) at a distance \[\overset{\to }{\mathop{r}}\,\]from element carrying current (i).
Biot-Savart law is used to compute the magnetic responses, even at a very small range. This law is similar to that of Coulomb’s law of electrostatics.
According to Biot-Savart law,
The small area of the magnetic field is directly proportional to the flow of current. \[dB\propto i\]
The small area of the change in a magnetic field is directly proportional to the vector distance. \[dB\propto r\]
The small area of the change in a magnetic field is directly proportional to the small current element. \[dB\propto dl\]
The small area of the change in a magnetic field is inversely proportional to the cube of the distance. \[dB\propto \dfrac{1}{{{r}^{3}}}\]

Combining all the above proportionality equations, we get, \[dB\propto i\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{{{r}^{3}}} \right)\]
To remove this proportionality constant, we use, \[\dfrac{{{\mu }_{0}}}{4\pi }\].
\[dB=\dfrac{{{\mu }_{0}}}{4\pi }i\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{{{r}^{3}}} \right)\]
The value of the constant, magnetic permeability of free space, is given to be, \[{{\mu }_{0}}=4\pi \times {{10}^{-7}}{H}/{m}\;\]
\[\therefore \] The magnetic field (dB) due to smaller element (dl) at a distance \[\overset{\to }{\mathop{r}}\,\]from element carrying current i, is \[dB=\dfrac{{{\mu }_{0}}}{4\pi }i\left( \dfrac{\overset{\to }{\mathop{dl}}\,\times \overset{\to }{\mathop{r}}\,}{{{r}^{3}}} \right)\]
So, the correct answer is “Option D”.

Note: The direction of the magnetic field is perpendicular to the plane containing the small current element and the distance. The magnetic field is directed inward. The current element is a vector quantity. Biot-Savart’s law gives the equation of the magnetic field produced due to the current carrying element.