Question
Explain Biot-Savart Law.
✓
Answer
→Law : "The intensity of magnetic field due to an electric current element I $\overrightarrow{d l}$ at a point having position vector $\vec{r}$ with respect to the electric current element is given by the formula, $d \overrightarrow{ B }=\frac{\mu_0}{4 \pi}=\frac{ I \overrightarrow{d l} \times \hat{ r }}{r^2}, \because$
→As shown in the figure, a finite conductor XY carries current I.
→A point P is located at some distance from the conductor. We want to find magnetic field at point P . For this, consider the conductor as divided in to an small elements and choose one small current element $I d \vec{l}$ from conductor.
→$\vec{r}$ is the position vector of point P from this current element $I d \vec{l}$. The angle between $I d \vec{l}$ and the position vector $\vec{r}$ is $\theta$.
→According to Biot-Savart's law, the magnitude of magnetic field $d \overrightarrow{ B }$ is proportional to the current I , the element length $\mid \vec{d} \vec{l}$ and inversely proportional to the square of the distance $r$. It's direction is perpendicular to the plane of $d \vec{l}$ and $\vec{r}$.
→The direction of the magnetic field can be found using the right hand screw rule.
→Vector form,
$\begin{array}{l}
\text { Vector form, } \\
\qquad d \overrightarrow{ B } \propto \frac{ I d \vec{l} \times \vec{r}}{r^3} \\
\therefore \quad d\overrightarrow{ B }=\frac{\mu_0}{4 \pi} \cdot \frac{ I d \vec{l} \times \vec{r}}{r^3}
\end{array}$
Where, $\frac{\mu_0}{4 \pi}$ is a constant of proportionality and its SI unit has the exact value,
$\frac{\mu_0}{4 \pi}=10^{-7} \frac{ Tm }{ A }$
$\mu_0=$ the permeability of free space.
Equation (1) holds when the medium is only vacuum.
→The magnitude of this field is :
$\begin{aligned}
d B & =\frac{\mu_0}{4 \pi} \cdot \frac{ I d l r \sin \theta}{r^3} \\
\therefore \quad d B & =\frac{\mu_0}{4 \pi} \cdot \frac{ I d l \sin \theta}{r^2}
\end{aligned}$
→To obtain the total magnetic field at point $P$ due to entire wire, equation (1) has to be integrated.
$\begin{aligned}
\overrightarrow{ B } & =\int d \overrightarrow{ B } \\
\overrightarrow{ B } & =\frac{\mu_0}{4 \pi} \int \frac{ I d \vec{l} \times \hat{r}}{r^2}
\end{aligned}$
→As shown in the figure, a finite conductor XY carries current I.
→A point P is located at some distance from the conductor. We want to find magnetic field at point P . For this, consider the conductor as divided in to an small elements and choose one small current element $I d \vec{l}$ from conductor.
→$\vec{r}$ is the position vector of point P from this current element $I d \vec{l}$. The angle between $I d \vec{l}$ and the position vector $\vec{r}$ is $\theta$.
→According to Biot-Savart's law, the magnitude of magnetic field $d \overrightarrow{ B }$ is proportional to the current I , the element length $\mid \vec{d} \vec{l}$ and inversely proportional to the square of the distance $r$. It's direction is perpendicular to the plane of $d \vec{l}$ and $\vec{r}$.
→The direction of the magnetic field can be found using the right hand screw rule.
→Vector form,
$\begin{array}{l}
\text { Vector form, } \\
\qquad d \overrightarrow{ B } \propto \frac{ I d \vec{l} \times \vec{r}}{r^3} \\
\therefore \quad d\overrightarrow{ B }=\frac{\mu_0}{4 \pi} \cdot \frac{ I d \vec{l} \times \vec{r}}{r^3}
\end{array}$
Where, $\frac{\mu_0}{4 \pi}$ is a constant of proportionality and its SI unit has the exact value,
$\frac{\mu_0}{4 \pi}=10^{-7} \frac{ Tm }{ A }$
$\mu_0=$ the permeability of free space.
Equation (1) holds when the medium is only vacuum.
→The magnitude of this field is :
$\begin{aligned}
d B & =\frac{\mu_0}{4 \pi} \cdot \frac{ I d l r \sin \theta}{r^3} \\
\therefore \quad d B & =\frac{\mu_0}{4 \pi} \cdot \frac{ I d l \sin \theta}{r^2}
\end{aligned}$
→To obtain the total magnetic field at point $P$ due to entire wire, equation (1) has to be integrated.
$\begin{aligned}
\overrightarrow{ B } & =\int d \overrightarrow{ B } \\
\overrightarrow{ B } & =\frac{\mu_0}{4 \pi} \int \frac{ I d \vec{l} \times \hat{r}}{r^2}
\end{aligned}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
An alternating voltage of frequency f is applied across a series LCR circuit. Let fr be the resonance frequency for the circuit. Will the current in the circuit lag, lead or remain in phase with the applied voltage when (i) f > f r, (ii) f < fr? Explain your answer in each case.
Explain the effect of dielectric on the capacitance of parallel plate capacitor and derive the equation of dielectric constant.
A filter transmits only the radiation of wavelength greater than 440nm. Radiation from a hydrogen-discharge tube goes through such a filter and is incident on a metal of work function 2.0eV. Find the stopping potential which can stop the photoelectrons.
By what percent will the capacitance of a spherical conductor increase if the amount of charge on a spherical conductor is tripled?
The susceptibility of annealed iron at saturation is 5500. Find the permeability of annealed iron at saturation.
Two large glass plates are placed vertically and parallel to each other inside a tank of water with separation between the plates equal to 1mm. Find the rise of water in the space between the plates. Surface tension of water = 0.075N/m.
(a) Calculate the potential at a point $P$ due to a charge of $4 \times 10^{-7} C$ located $9 cm$ away.
(b) Hence obtain the work done in bringing a charge of $2 \times 10^{-9} C$ from infinity to the point P. Does the answer depend on the path along which the charge is brought?
→(b) Hence obtain the work done in bringing a charge of $2 \times 10^{-9} C$ from infinity to the point P. Does the answer depend on the path along which the charge is brought?
Show diagrammatically when is magnetic flux taken as:
- Positive.
- Negative.
A barometer tube is 80cm long (above the mercury reservoir). It reads 76cm on a particular day. A small amount of water is introduced in the tube and the reading drops to 75.4cm. Find the relative humidity in the space above the mercury column if the saturation vapour pressure at the room temperature is 1.0cm.
When a boron nucleus $\big(\text{ }^{10}_5\text{B}\big)$ is bombarded by a neutron, an $\alpha$-particle is emitted. Which nucleus will be formed as a result?
→