Biot-Savart Law vs Coulomb’s Law

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Electricity and magnetism are two fundamental branches of physics that explain how charges and currents interact with the physical world. Two important laws—Coulomb’s Law and Biot-Savart Law play a central role in understanding electric and magnetic fields, respectively.
Biot-Savart Law-Magnetic fields
Magnetic fields
While Coulomb’s Law deals with forces and fields produced by stationary electric charges where the Biot-Savart Law describes the magnetic field created by moving charges or electric currents.
Although both laws share mathematical similarities, they apply to different situations and reveal deeper relationships between electricity and magnetism. This note explains both laws in detail and highlights their differences.
COULOMB’S LAW
Definition
Coulomb’s Law gives the quantitative measure of the electrostatic force between two stationary charges. It states that:
The electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Mathematical Form F = 1 / 4Π€0 . q1q2 /r2
Where:
F = electrostatic force
q1 and q2 = magnitudes of charges
r = distance between charges
€0 = permittivity of free space
Nature of Force
The force is central (acts along the line joining the charges).
It can be attractive or repulsive:
Like charges repel.
Unlike charges attract.
Electric Field from Coulomb’s Law
By modifying the law, we obtain the electric field due to a point charge: E = 1 / 4Π€0 . q /r2
The electric field exists whether the charge is moving or stationary, but Coulomb’s Law itself applies only when charges are stationary (electrostatic case).
Applications
Calculating forces in electrostatics
Derivation of Gauss’s Law for point charges atomic structure, capacitors, and charge distribution problems
Biot-Savart Law-Capacitors
Capacitors

BIOT-SAVART LAW

Definition
Biot-Savart Law provides the magnetic field produced at a point due to a current-carrying conductor. It applies to moving charges because electric current is a flow of charged particles.
The law states that: The magnetic field dB at a point due to a small current element ( I dl} ) is directly proportional to the current, length of the element and sine of the angle between element and the line joining the point and inversely proportional to the square of the distance from the element.
Mathematical Form
dB = µ0 / 4Π . Idl x r / r2
Where:
dB = small magnetic field magnitude
I  = current through the conductor
dl  = small length element of wire
r  = distance of point from the element
µ0 = permeability of free space
The direction of dB is given by the right-hand thumb rule.
Magnetic Field from a Long Straight Conductor
Integrating the Biot-Savart Law gives: B = µ0 I / 4Πr

Nature of Magnetic Field

Magnetic field is vector in nature.
It exists only due to moving charges or current.
Magnetic fields are always perpendicular to the plane formed by the current element and position vector.

Applications

Magnetic field of straight wires, circular coils, and solenoids
Foundation of Ampere’s Law
Used in designing motors, electromagnets, inductors, transformers, etc.

SIMILARITIES BETWEEN THE TWO LAWS

Inverse-Square Relation
Both Coulomb’s Law and Biot-Savart Law contain a term proportional to ( 1/r2} ), showing that both electric and magnetic effects reduce rapidly with distance.
Field Calculation
Coulomb’s Law calculates electric field due to charges.
Biot-Savart Law calculates magnetic field due to currents.
Superposition Principle
Fields from multiple charges or current elements can be added vectorially.

Vector Nature

Both laws deal with vector quantities and require direction as well as magnitude.

DIFFERENCES BETWEEN COULOMB’S LAW AND BIOT-SAVART LAW:-                                                       

Point of Comparison

Coulomb’s Law          

Biot-Savart Law

Physical Quantity  

Force and electric field      

Magnetic field                                                                       

Source             

Stationary electric charges   

Moving charges (current)                                                             

Type of Interaction

Electrostatic                                                                                   

Magnetostatic   

   

Nature of Force    

 

Can be attractive or repulsive  

Magnetic field does not “attract or repel”; it influences moving charges and currents.

Dependence         

Scalar product of charges     

Vector cross product involving angle                                                 

Applicability

Only static charges             

Currents steady in time   

RELATIONSHIP IN ELECTROMAGNETISM

Although the two laws appear different, they complement each other in showing that:
Electric charges produce electric fields.
Moving charges produce magnetic fields.
Together, they help form the mathematical foundation of Maxwell’s Equations, which unify electricity and magnetism into a single framework called Electromagnetism.

CONCLUSION

Coulomb’s Law and Biot-Savart Law are cornerstone principles in the study of electrostatics and magnetostatics. Coulomb’s Law quantifies interactions between stationary charges, while the Biot-Savart Law defines magnetism produced by moving charges or currents.
Both show inverse-square dependence and vector behaviour, but their domains are distinct. Understanding these laws not only enables calculation of electric and magnetic fields in various situations but also provides deeper insight into the unified nature of electromagnetic phenomena.
Coulomb’s Law explains the force or electric field produced by stationary electric charges, whereas Biot–Savart Law describes the magnetic field produced by moving charges or electric current.
 
Coulomb’s Law is used to calculate the electric field, which is produced by electric charges at rest (static charges).
 
 
Biot–Savart Law is used to calculate the magnetic field, produced by an electric current or moving charge element.
 
In Coulomb’s Law, the effect depends on the magnitude of electric charge and the distance between the charges.
 
In Biot–Savart Law, the effect depends on the current element (Idl), the distance from the observation point, and the angle between the current direction and the position vector.
 
Yes. Both Coulomb’s Law and Biot–Savart Law follow the inverse-square relationship, meaning the field strength decreases with the square of the distance from the source.
 
No. Coulomb’s Law strictly applies to static (rest) charges, not moving charges. For moving charges or currents, laws like Biot–Savart Law and Ampère’s Law are used.

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