State Biot Savart Law
The magnetic field created by a constant electric field is described by the Biot Savart Law equation. According to this law, the magnitude, length, direction, and proximity of the electric current is related to the magnetic field. Also, this law is consistent with both Gauss's theorem and Ampere's circuital law. As Coulomb's law is fundamental for electrostatics, similarly, Biot Savart Law is also fundamental for magnetostatics. In 1820 two French physicists Jean Baptiste Biot and Felix Savart created the Biot Savart Law statement, which derived the mathematical formula for magnetic flux density at a point due to a nearby current-carrying conductor. Witnessing the deflection of the magnetic compass needle, these scientists concluded that the current particle projects a magnetic field into the space surrounding it.
The mathematical expression was derived through observation and calculations, which showed the magnetic flux density. This flux density dB is proportional to the length of the current I, element dl, the sine of angle θ between the direction of the current and the vector joining a given point of a magnetic field and the current element. It is inversely proportional to the square of the distance of the given point from the current element r.
Biot Savart Law Derivation
The biot savart law formula can be given as
\[dB\ \alpha\frac{Idl\sin\theta}{r^{2}}\]
Or
\[dB=k\frac{Idl\sin\theta}{r^{2}}\]
Where, k is constant, depending upon the magnetic properties of the medium and system of the units employed. In the SI system of the unit,
\[k = \frac {\mu_0 \mu_r}{4\pi}\]
The final Biot-Savart law derivation is expressed as,
\[db = \frac {\mu_0 \mu_r}{4\pi}\times \frac{Idl\sin\theta}{r^{2}}\]
Consider a long wire carrying current I and at a point p in space. The wire is represented in the picture below by red color. Also, consider an infinitely small length of the wire dl at a distance r from the point P as shown in the diagram. Here, r is the distance vector, which makes the angle θ with the direction of the current in the minute portion of the wire.
Imagining the condition, you can understand that the magnetic field density at point P due to the minute length dl of the wire is proportional to the current carried by this portion of the wire. As the current through the minute length of the wire is similar to the current carried by the whole wire itself, we can express it as:
dB ∝ I
It is also natural to think that the magnetic field density at point P due to the minute length dl of wire is inversely proportional to the square of the straight distance from Point P to the center of dl. Mathematically it can be expressed as:
dB ∝ \[\frac{1}{r^{2}}\]
Also, the magnetic field density at the point P due to the minute portion of the wire is proportional to the actual length of the minute length dl of wire. θ is the angle between distance vector r and direction of current through this minute portion of the wire, the component dl of the wire facing perpendicular to the point P is dlsinθ.
Hence, dB ∝ dl sinθ
Now, merging these three statements, we can write,
\[dB\ \alpha k\frac{I.dl.sin\theta}{r^{2}}\]
The formula for Biot-savart law can be stated here as follows:
Now, introducing the value of constant k (which we have already mentioned at the starting of this derivation) in the above formula, we get,
\[dB = \frac{kl.dl.\sin\theta}{r^{2}}\]
\[ K = \frac{\mu_{0}\mu_{r}}{4\pi}\]
Here, μ0 is the absolute permeability of air or vacuum used in the expression of constant k, and its value is 4π10-7 Wb/ A-m in the SI system of units. μr in the expression of constant k is the relative permeability of the medium.
Now, flux density (B) at point P due to the total length of the current-carrying conductor or wire can be expressed as,
\[B = \int dB = dB= \int\frac{\mu_{0}\mu_{r}}{4\pi}\times\frac{kl.dl.\sin\theta}{r^{2}}\]
\[= \frac{\mu_{0}\mu_{r}}{4\pi}I \int\frac{\sin\theta. dl}{r^{2}}\]
If D is the perpendicular distance of point P from the wire, then
r sinθ = D or r =\[\frac{D}{\sin\theta}\]
Now, the formula of flux density B at point P can be rewritten as,
\[B = \frac {I\mu_0 \mu_r}{4\pi}\int \frac{\sin\theta}{r^{2}}dl = \frac {I\mu_0 \mu_r}{4\pi}\int\frac{\sin^{3}\theta}{D^{2}}dl\]
Again, \[\frac{l}{D}\] = cotθ
⇒l = D cotθ
Therefore, dl = -Dcosec2θdθ
Finally, the expression of B comes as,
\[B = \frac {I\mu_0 \mu_r}{4\pi}\int \frac{\sin^{3}\theta}{D^{2}} [-Dcosec^{2}\theta d\theta\]
=\[-\frac {I\mu_0 \mu_r}{4\pi D}\int \sin^{3}cosec^{2}\theta d\theta\]
=\[-\frac {I\mu_0 \mu_r}{4\pi D}\int \sin\theta d\theta\]
This angle θ is dependent upon the length of the wire and position of point P. Say for a certain partial length of the wire, angle θ, as indicated in the figure above, varies from θ1 to θ2. Hence, magnetic flux density at the point P due to the total length of the conductor is,
\[B = -\frac {I\mu_0 \mu_r}{4\pi D}\int_{\theta 1}^{\theta 2}\sin\theta d\theta\]
= \[-\frac {I\mu_0 \mu_r}{4\pi D}[-\cos\theta]_\theta1^\theta2\]
= \[\frac {I\mu_0 \mu_r}{4\pi D} [\cos\theta_{1}- \cos\theta_{2}]\]
Let's consider the wire is infinitely long, then θ will change from 0 to π that is θ1 = 0 to θ2 = π.
Placing these two values in the above final expression of Biot Savart law, we get,
\[B = \frac {I\mu_0 \mu_r}{4\pi D}[\cos0 - \cos\pi = \frac {I\mu_0 \mu_r}{4\pi D} 1-(-1)\]
\[B = \frac{\mu_{0}\mu_{r}I}{2\pi D}\]
Applications of Biot Savart Law
The applications of Biot Savart Law are mentioned below.
The Biot Savart Law is used to calculate magnetic responses at the molecular and atomic levels.
The Biot Savart Law is used in aerodynamic theory while calculating velocity generated by vortex lines.
Importance of Biot Savart Law
The Biot Savart Law is important for the following reasons:
The Biot Savart law can be applied to small conductors that carry current.
This law can also be applied to symmetrical current distribution.
FAQs on Biot Savart Law - Statement, Derivation and Applications
1. What is the Biot-Savart Law and what is its mathematical expression in vector form for JEE Main 2026?
The Biot-Savart Law is a fundamental principle in magnetostatics used to calculate the magnetic field generated by a constant electric current. It states that the magnetic field (dB) produced by an infinitesimal current element (I dl) at a point is directly proportional to the current, the length of the element, and the sine of the angle between the element and the position vector (r), and inversely proportional to the square of the distance from the element. In vector form, which is crucial for solving JEE problems, the expression is:
dB = (μ₀/4π) * (I dl × r̂) / r²
Where μ₀ is the permeability of free space, I is the current, dl is the current element vector, r̂ is the unit vector pointing from the element to the point, and r is the distance.
2. How is the Biot-Savart Law applied to derive the magnetic field at the centre of a circular current-carrying loop?
To find the magnetic field at the centre of a circular loop of radius R carrying current I, we apply the Biot-Savart Law to a small element dl on the loop. For any element dl, the position vector r to the centre is always perpendicular to it (θ = 90°, so sin θ = 1). The direction of the magnetic field dB from each element points along the axis (by the right-hand rule) and they all add up. Integrating dB = (μ₀/4π) * (I dl / R²) over the entire circumference (∫dl = 2πR), we get the total magnetic field:
B = (μ₀I / 4πR²) * ∫dl = (μ₀I / 4πR²) * (2πR)
Therefore, the final expression is B = μ₀I / 2R.
3. What is the expression for the magnetic field on the axis of a circular current loop using Biot-Savart's law?
For a point on the axis of a circular loop of radius R at a distance x from its centre, the magnetic field can be derived using the Biot-Savart Law. The components of the magnetic field perpendicular to the axis cancel out due to symmetry, while the components along the axis add up. The final expression for the magnitude of the magnetic field (B) on the axis is:
B = (μ₀IR²) / 2(x² + R²)^(3/2)
This is a high-yield formula for JEE Main and Advanced, often used in problems involving magnetic fields of rings and coils.
4. For JEE problem-solving, when is it better to use Ampere's Law instead of the Biot-Savart Law?
The choice between Biot-Savart Law and Ampere's Circuital Law depends on the symmetry of the current distribution. You should use:
- Ampere's Law for situations with high symmetry where you can define a suitable Amperian loop. This includes calculating the magnetic field of an infinitely long straight wire, a solenoid, or a toroid. It simplifies the calculation significantly.
- Biot-Savart Law for cases with low or no symmetry. It is essential for finding the magnetic field of finite-length wires, circular arcs, or any arbitrary shape of current-carrying conductor, as it calculates the contribution from each infinitesimal current element.
5. What are the key limitations of the Biot-Savart Law that a JEE aspirant should be aware of?
While powerful, the Biot-Savart Law has certain limitations:
- It is only applicable for steady currents (direct currents or DC) that do not change with time. It is a law of magnetostatics.
- It is analogous to Coulomb's Law but is more complex due to the vector cross product, which makes calculations for non-symmetric systems computationally intensive.
- The law deals with an infinitesimal current element (Idl), which cannot exist in isolation. It must be integrated over a complete circuit to find the total magnetic field.
6. How does the Biot-Savart Law draw a parallel to Coulomb's Law in electrostatics?
Biot-Savart Law in magnetostatics is often considered the analogue of Coulomb's Law in electrostatics. Here’s a comparison:
- Source: The source of the electric field in Coulomb's law is a scalar point charge (q), whereas the source of the magnetic field in Biot-Savart law is a vector current element (I dl).
- Distance Dependence: Both laws follow an inverse square law, where the field strength is proportional to 1/r².
- Direction: The electrostatic field (E) is directed along the position vector connecting the source and the point. The magnetostatic field (B) is directed perpendicular to both the current element (dl) and the position vector (r), as defined by the cross product.
Understanding this analogy helps in remembering the core concepts of both fields.
7. Can the Biot-Savart Law be derived from Maxwell's Equations?
Yes, the Biot-Savart Law can be derived from Maxwell's Equations under magnetostatic conditions. Starting with Ampere's Law (∇ × B = μ₀J) and the definition of the magnetic vector potential (B = ∇ × A), one can arrive at a Poisson-like equation for A (∇²A = -μ₀J) under the Coulomb gauge (∇ · A = 0). The solution to this equation for the vector potential A leads directly to the integral form of the Biot-Savart Law. This shows that the law is consistent with the more fundamental framework of Maxwell's Equations.
