Biot-Savart’s Law | Laplace’s Law

Biot Savart’s law is experiment done by Biot and Savart to find magnetic field induction at a point due to small current element.

In 1820 Oersted found that when current in passes through a conductor, magnetic field is produced around it. Just at the same time, Laplace gave a rule for calculation magnitude of magnetic field produced. It is known as Laplace’s law or Biot-Savart’s law.

Derivation of Biot Savart’s Law

Consider a conductor XY through which current I is flowing. Due to it, magnetic field is produced around it. To find out magnetic field produced at point P, let us consider as element AB of length dl which make angle θ with line joining r as shown in figure 1. Let the distance of P from dl be r.

biot-and-savart-law

 

According to the Biot-Savart’s Law, magnetic field strength produced by the element at that point P is

  1. Directly proportional to magnitude of current passed i.e. dB\propto I
  2. Directly proportional to length of the element i.e. dB\propto dl
  3. Directly proportional to sine of the angle between the conductor and the line joining r i.e. dB\propto sin\theta
  4. Inversely proportional to square of the line joining between center of the element and the point P i.edB\propto \dfrac{1}{r^2}

Combining these factor, we get

dB\alpha\dfrac{I dl sin \theta}{r^2} dB = \dfrac{k I dl sin \theta}{r^2}

Where k is proportionality constant and its value is dependent upon the system of units chosen for measurement of the various quantities.

In SI units, K = \dfrac{\mu_0}{4\pi} and In cgs system K = 1.

Where \mu_0 is absolutely magnetic permeability of free space

And \mu_0 = 4\pi 10^{-7}Wb A^{-1}m^{-1} = 4\pi 10^{-7}T A^{-1}m

( 1T = 1WB m-2)

In SI units, dB = \dfrac{\mu_0}{4\pi}\dfrac{Idl sin\theta}{r^2}——–(i)

In cgs system, dB = \dfrac{Idl sin\theta}{r^2}

In vector form, we may write

|d\vec{B}| = \dfrac{\mu_0}{4\pi}\dfrac{I|d\vec{l}\times\vec{r}|}{r^3}

Or,

d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I(d\vec{l}\times\vec{r})}{r^3}——–(ii)

Direction of d\vec{B}.

From equation (ii), the direction of d\vec{B} would be the direction of the cross product vector, d\vec{l}\times\vec{r}. It is represented by the Right handed screw rule or Right Hand Rule. Here d\vec{B} is perpendicular to the plane containing d\vec{l} and \vec{r} and is directed inwards. If the point P is to the left of the current element, d\vec{B} will be perpendicular to the plane containing d\vec{l}and \vec{r} directed outwards.

Magnetic field induction at point P due to current through entire wire is

 \vec(B) = \int\dfrac{\mu_0}{4\pi}\dfrac{I d\vec{l}\times\vec{r}}{r^3} B = \int\dfrac{\mu_0}{4\pi}\dfrac{I dl sin\theta}{r^2}

Biot Savart’s law in term of current density J, states that

d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{\vec{J}\times\vec{r}}{r^3}dV  (J = \dfrac{1}{A} = \dfrac{I dl}{A dl} = \dfrac{I dl}{dV})

where J = current density at any point on the current element, dV = volume of the element.

Biot Savart’s law in terms of charge (q) and its velocity (v) is

d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{\vec{v}\times\vec{r}}{r^3}dV (I d\vec{l} = \dfrac{q}{dt}.d\vec{l} = q\dfrac{d\vec{l}}{dt} = q\vec{v} )

Biot Savart’s law in term of magnetising force or magnetising (H) of the magnetic field s:

In SI or mks system

d\vec{H} = \dfrac{d\vec{B}}{\mu_0} = \dfrac{1}{4\pi}\dfrac{I d\vec{l}\times\vec{r}}{r^3} = \dfrac{1}{4\pi}\dfrac{I d\vec{l}\times\vec{r}}{r^2}

and

dH = \dfrac{1}{4\pi}\dfrac{I dl sin\theta}{r^2}

In cgs e.m. units

d\vec{H} = \dfrac{Id\vec{l}\times\vec{r}}{r^3}

And

dH = \dfrac{I dl sin\theta}{r^2}

Some important features of Biot Savart’s law

  1. Biot Savart’s law is valid for a symmetrical current distribution.
  2. Biot Savart’s law is applicable only to very small length conductor carrying current.
  3. This law cannot be easily verified experimentally as the current carrying conductor of very small length cannot be obtained.
  4. This law is analogous to Coulomb’s law in electrostatics.
  5. The direction of d\vec{B} is perpendicular to both I d\vec{l} and \vec{r}.
  6. If  \theta = 0^0 i.e. the point P lies on the axis of the linear conductor carrying (or on the wire carrying current) then dB = \dfrac{\mu_0}{4\pi}\dfrac{I dl sin 0^0}{r^2} = 0. It means there is no magnetic field induction at any point on the thin linear current carrying conductor.
  7. If \theta = 90^0 i.e. the point P lies at a perpendicular position w.r.t., current element, then
    dB = \dfrac{\mu_0}{4\pi}\dfrac{I dl}{r^2}, which is maximum.
  8. If  \theta = 90^0 or 180^0 , then dB = 0 i.e. minimum.

 

SIMILARITIES AND DIFFERENCES BETWEEN THE BIOT-SAVART’S LAW FOR THE MAGNETIC FIELD AND COULOMB’S LAW FOR ELECTROSTATIC FIELD

Similarities

  1. Both the laws for fields are long range, since in both the laws, the field at a point vary inversely as the square of the distance from the source to point of observation.
  2. Both the fields obey superposition principle.
  3. The magnetic field is linear in the source I d\vec{l}, just as the electric field is linear in its source, the electric charge q.

Dis-similarities

  1. The electrostatic field is produced by a scalar source namely, the electric charge q and the magnetic field is produced by a vector source, a current element Id\vec{l}.
  2. The electrostatic field is acting along the displacement vector, i.e., line joining the source and the field point. The magnetic field is acting perpendicular to the plane containing the current element Id\vec{l} and displacement vector \vec{r}, i.e., along the direction(Id\vec{l}\times\vec{r}).
  3. The electrostatic field at a point due to a charge is independent of angle \theta, whereas the magnetic field at a point due to a current element is angle dependent. It means Coulomb law is independent of angle whereas the Biot-Savart’s law is angle dependent.

2 Thoughts to “Biot-Savart’s Law | Laplace’s Law”

  1. Sai Harsha

    Is it mandatory to use only 4ohm loud speaker or can i use any other

  2. Priyanka Shankholia

    how to calculate the values of components????like how you know whcih resistor and transistors are to be used????

Comments are closed.