Magnetic Field Due to a Current Element: Biot Savart Law

Magnetic field due to current element is given by Biot-Savart Law. Biot-Savart Law states that if a current carrying conductor of length dl produces a magnetic field dB, the force on another similar current carrying conductor depends upon the size, orientation and length of the first current carrying element. Biot-Savart law establishes a relation between the magnitude, direction, length and proximity of the magnetic field to the current element. Here, the current element is taken as a vector quantity. 

Read More: Moving Charges and Magnetism Important Questions

Key Takeaways: Biot-Savart Law, Electric Field, Magnetic Field, Moving Charges, Magnetism, Vector Quantity, Coulomb’s Law.

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Moving Charges and Magnetism

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Stationary electric charges exert a force on other stationary chargers near them. This force is approximated using Coulomb’s Law, which states that: The force on particle q₁ by q₀ at x₁ distance is given by (r_1,0, a unit vector in direction of x₀ to x­₁)

\(F = \frac{1}{4} \pi \epsilon_0 \frac{q_1 q_0 }{(x_1 - x_0)^2} \hat{r}_{1,0}\)

Thus, the field generated by q₀ around it is given by-

\(E(x_0) = \frac{F}{q_0} = \frac{1}{4} \pi \epsilon_0 \frac{q_1}{(x_1 - x_0)^2} \hat{r}_{1,0}\)

Interestingly moving charges also exert a force around themselves and this force is magnetic in nature. This effect of moving charges (or current) was accidentally discovered by Hans Christian Oersted, who noticed that a current-carrying wire was causing the needle of the nearby placed compass to deflect.

Subsequent experimentation established that the deflection of the needle was always tangential to an imaginary circle with its plane perpendicular to the wire. Subsequently, Iron filling experimentation demonstrated that indeed the wire was producing concentric magnetic fields perpendicular to it.

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Biot-Savart Law

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Biot-Savart Law states that if a current-carrying conductor of length dl produces a magnetic field dB, the force on another similar current-carrying conductor depends upon the size, orientation, proximity and length of the first current carrying element. Just like Coulomb's Law was used to approximate the Electric field generated by a stationary charge, Biot-Savart's Law is used to approximate the magnetic field generated by the current.

Assume that a conductor of a very large length L is carrying current I through it. The magnetic field due to the current, B is perpendicular to the plane of the conductor. Further, let us assume that a section of this conductor, say dL is producing a section of the magnetic field dB at point r away from it in the same plane. Let the angle between dL and dB in the direction of r be Θ.

Biot-Savart Law Video Explanation

Then Biot-Savart Law states that,

• dB is proportional to the current I
• dB is proportional to element length dL
• dB is inversely proportional to the square of distance r
• dB has a direction perpendicular to the plane containing r and dL.

\(dB \propto Idl.r/r^3\)

\(\frac{\mu_0}{4\pi} Idl.r/r^3\)

Where µ₀/4∏ is a constant (µ ₀ is ‘permeability of free space’) and it is equal to 10^-7 Tm/A. Note that the above equation holds true in vacuum only

Notice that the above equation is in vector form i.e., vector dB has both magnitude and direction. In order to find only the magnitude of DB.

\(|dB| =\frac{\mu_0}{4\pi} Idl.\frac{sin \theta}{r^2}\)

Direction of Magnetic Field

Since the magnetic field is a result of the cross product of dL and r the resultant will have the direction as appropriated by Fleming’s right-hand rule.

Where a is the direction of the current, b is the direction of vector r and a x b is the direction of the magnetic field

Read More: Lorentz Force

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Unit of Magnetic Field

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B is also called Magnetic Flux Density and is measured in Tesla (T)

The SI base units of Tesla are ‘Kilogram per the Second square per Ampere’  or Kgs^-2A^-1

The dimensional formula for Magnetic Flux Density is given by [M¹ T^-2 A^-1]

Where, 

M= Mass (Kg)

T= Time (seconds ‘s’)

A= Ampere (Unit of Electricity)

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Magnetic Field on the axis of Current Carrying Circular Loop

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Consider a loop of radius R carrying current I through it and we wish to calculate the magnetic field generated by it at the point P from the loop whose center is assumed to be at the origin θ.

Now, the magnitude of magnetic force at P which is x distance away from O and r from element dL is given by

\(dB =\frac{\mu_0}{4\pi}\frac{ |Idl.r|}{r^2}\)

Now by the right-angle triangle rule, r² = R² + x^2, and each element of the loop would be perpendicular to the direction vector r since clearly, dL lies in the Z-Y axis whereas r lies in the X-Y axis.

So, | dL x r | = r.dL ( sin 90 =1)

Now the equation becomes,

\(dB =\frac{\mu_0}{4\pi}\frac{ Idl.r}{(x^2 + R^2)}\)

The magnetic field dB has a direction perpendicular to the dL and r plane, which can be √ broken into two components B_x along X-axis and dB⊥ which along the axis perpendicular to X-axis.

The cosine representation of the angle θ is given by,

\(cos \theta=\frac{ R}{(x^2 + R^2)^{1/2}}\)

Now dB⊥ would be in one direction for the upper half of the loop and in the other direction for the lower half as the motion of current is in opposite directions in the upper and lower half of the loop (due to its anticlockwise/ clockwise) motion.

Since the loop is symmetrical about the original the net dB⊥ on the point P would be zero.

dB⊥ = 0

Now the magnetic field applied solely depend upon B_x

\(dB_x=\frac{\mu_0 Idl}{\frac{4\pi.R}{(x^2 + R^2)^{3/2}}}\)

Now, dL when summed up would be equal to the circumference of the loop 2∏R

By this, the magnetic field on P by the whole loop would be-

\(B =B \times \hat{i} =\frac{ \mu_0 IR^2}{ 2(x^2 + R^2)^{\frac{3}{2}}}.\hat{i}\)

If magnetic field along the center of the loop (at O) has to be found out, then x² ­= 0 and the equation becomes –

\(B_0 = \frac{ \mu_0 I}{2 R.}\hat{i}\)

Read More: Electromagnetic Waves

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Difference between Electric and Magnetic Field

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Key differences between Electric and Magnetic field are tabulated below.

Read more: Electromagnetic Induction

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Things to Remember

• The magnetic field produced by a current carrying conductor in a specific direction can be determined using Biot-Savart Law.
• Biot-Savart Law states that if a current carrying conductor of length dl produces a magnetic field dB, the force on another similar current carrying conductor depends upon the size, orientation and length of the first current carrying element.
• Magnetic Flux Density and is measured in Tesla (T) or Kgs^-2A^-1
• The dimensional formula for Magnetic flux density is [M¹ T^-2 A^-1].
• Current element is taken as a vector quantity. 

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