Biot-savart Law - Practice Questions & MCQ

Biot-Savart Law:-

• If a point charge q is kept at rest near a current carrying wire, It is found that no force acts on charge. It means a current carrying wire does not produce electric field.

• However, if the charge q is projected in the direction of the current with velocity v, then it is deflected towards the wire (q is assumed positive). There must be a field at P which exerts a force on the charge when it is projected, but not when it is kept at rest. This field is different from the electric field which always exerts a force on a charged particle whether it is at rest or in motion. This new field is called magnetic field and is denoted by the symbol B. The force exerted by a magnetic field is called magnetic force.

According to Biot Savart's Law the magnetic induction dB at point P due to the elemental wire segment AB as shown in the figure depends upon four factors which are given as

(i) dB is directly proportional to the current in the element.

$
d B \propto I
$

(ii) dB is directly proportional to the length of the element

$
d B \propto d l
$

(iii) dB is inversely proportional to the square of the distance r of the point P from the element

$
d B \propto \frac{1}{r^2}
$

Combining above factors, we have
$d B \propto \frac{I d l \sin \theta}{r^2}$

$
d B=K \frac{I d l \sin \theta}{r^2}
$

Where K is a proportionality constant and its value depends upon the nature of the medium surrounding the current carrying wire. Its SI Units its value is given as

$
K=\frac{\mu_0}{4 \pi}=10^{-7} \mathrm{~T}-\mathrm{m} / \mathrm{A}
$
 

here, i is the current, $d \vec{l}$ is the length-vector of the current element and $\vec{r}$ is the vector joining the current element to the point P and $\theta$ is the angle between $d \vec{l}$ and $\vec{r}$.
$\mu_0$ is called the permeability of vacuum or free space. Its value is $4 \pi \times 10^{-7} \mathrm{~T}-\mathrm{m} / \mathrm{A}$.
The magnetic field at a point $P$, due to a current element in vacuum, is given by:
Vector form: $d \vec{B}=\frac{\mu_0}{4 \pi} \frac{(i d \vec{l} \times \vec{r})}{r^3}$
$\underline{\text { Scalar form: }} d B=\frac{\mu_0}{4 \pi} \frac{i d l \sin \theta}{r^2}$
For medium other than vacuum, $\mu_0$ will be replaced by $\mu$

$
\mu=\mu_0 \times \mu_r
$

where, $\mu_r$ is the relative permeability of the medium (also known as the diamagnetic constant of the medium)

Direction of magnetic field:

1. The rule of cross product

The direction of the field is perpendicular to the plane containing the current element and the point P according to the rules of cross product. If we place the stretched right-hand palm along $d l$ in such a way that the fingers curl towards $\vec{r}$, the cross product $d l \times \vec{r}$ is along the thumb Usually, the plane of the diagram contains both $d \vec{l}$ and $\vec{r}$. The magnetic field $d \vec{B}$ is then perpendicular to the plane of the diagram, either going into the plane or coming out of the plane. We denote the direction going into the plane by an encircled cross and the direction coming out of the plane by an encircled dot.

2. Right hand thumb rule

The direction of this magnetic induction is given by right hand thumb rule stated as "Hold the current carrying conductor in the palm of the right hand so that the thumb points in the direction of the flow of current, then the direction in which the fingers curl, gives the direction of magnetic field lines"

Cases:

 

Case 1. If the current is in a clockwise direction then the direction of the magnetic field is away from the observer or perpendicular inwards.

Case 2. If the current is in an anti-clockwise direction then the direction of the magnetic field is towards the observer or perpendicular outwards