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# SIMILARITIES AND DIFFERENCE BETWEEN BIOT-SAVART LAW AND COULOMB’S LAW

In this article, we are going to discuss about similarities and differences between Biot-Savart Law and **coulomb’s law**. So let’s start…

Both laws are used to measure the different quantities, one measures the electrostatic force then the other measures the magnetic field. Coulomb’s law measures the forces between two **static charges** whereas Biot-Savart Law is used to measure the magnetic field when the charges are in motion.

So before listing similarities and differences, let’s take a brief snapshots about coulomb’s law and Biot-Savart Law. [latexpage]

## COULOMB’S LAW

It is a very important law of **electrostatic**. It is analogy to **universal law of gravitation**. Coulomb’s law tells that how does two electric charges interact with each other and on what factors their interactions depends upon. If there is two stationary charges $q_1$ and $q_2$ seperated by the distance $r$, then

According to Coulomb’s Law-

- Electrostatic force is directly proportional to product of two charges i.e $\displaystyle{F\propto q_1 q_2}$.
**Electrostatic force**is inversely proportional to square of distance between two charges i.e $\displaystyle{F\propto\frac{1}{r^2}}$.- Combining all, we get $\displaystyle{F\propto\frac{q_1 q_2}{r^2}}$.
- Removing proportionality sign we get $\displaystyle{F=k\frac{q_1 q_2}{r^2}}$, where k is called electrostatic constant whose value in vaccum is $\displaystyle{k=8.85\times 10^{-12}C^2/Nm^2}$.

## BIOT-SAVART LAW

It is a very important law of **electrodynamics**. It is analogy to coulomb’s law. This law tells about the magnetic field at a point from the current carrying conductor and about the factors on which this **magnetic field** depends upon. According to Biot-Savart Law.

If a conductor is carrying current $I$ then the magnetic field at a point P due to infinitesimal portion of wire $dl$ will depends upon the following factors.

- $\displaystyle{dB\propto I}$
- $\displaystyle{dB\propto dl}$
- $\displaystyle{dB\propto sin\theta}$
- $\displaystyle{dB\propto \frac{1}{r^2}}$
- Combining all, we get $\displaystyle{dB \propto\frac{Idl\;sin\theta}{r^2}}$
- After removing proportionality sign, we get- $\displaystyle{dB= k\frac{Idl\;sin\theta}{r^2}}$, where k is called magnetic constant, whose value is $\displaystyle{10^{-7}N/A^2}$

So, after recalling coulomb’s law and Biot-Savart Law. Now let’s talk about their similarities and differences.

## SIMILARITIES BETWEEN COULOMB’S LAW AND BIOT-SAVART LAW

Some similarities between coulomb’s law and Biot-Savart Law is listed below:

- Both the electric and magnetic field follows the inverse square law.
- Both the electric and magnetic field have the long-ranges effect.
- Both the electric and magnetic field obey the
**Superposition Principle**. - Both the electric and magnetic field have the linear source, magnetic field depends upon $Idl$ (current element), whereas electric field depends upon q (electric charges).

## DIFFERENCE BETWEEN COULOMB’S LAW AND BIOT-SAVART LAW

Some differences between them is listed below:

- The
**electric charge**element $dq$ producing electric field is a scalar source whereas the electric current element $Idl$ producing magnetic field is a vector source having direction as directed by the right hand thumb rule. - In
**Biot-Savart Law**, the value of magnetic field depends upon the sine of angle θ between the distance vector r and direction of the current in infinitesimal portion $Idl$, but there is no such angle dependency for electric field. - Electric field is always directed along the
**displacement vector**joining the source and the point at which electric field is to be calculated, but the magnetic field is always perpendicular to the distance vector $r$ and the current element $Idl$ i.e cross product between $dl\times r$ gives the direction of magnetic field. - The relationship between the constant involved in the coulomb’s law and Biot-Savart Law is given as- $$\mu_0 \varepsilon_0=\frac{1}{c^2}$$ where, $\mu_0$ is the
**absolute permeability of the free****space**, $\varepsilon_0$ is the**absolute permittivity of the free space**and**c**is the**speed of light**in vaccum.

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