Biot-Savart Law with Explanation and Derivation, as detailed in Chapter Four of the NCERT textbook "MOVING CHARGES AND MAGNETISM" for Class 12 Physics (New Edition), is crucial for students preparing for competitive exams like JEE, and NEET. This topic is covered comprehensively in NCERT and RBSE curricula.
Introduction to Magnetic Fields
There are two reasons for all magnetic fields that we know
☆ Currents (or moving charges)
☆ Intrinsic magnetic moments of particles
Relationship Between Current and Magnetic Field
We know that Orsted's experiment, conducted by Hans Christian Orsted in 1820, was pivotal in establishing the relationship between electricity and magnetism.
Here, we will study the relation between current and the magnetic field it produces. It is given by Biot-Savart's law.
Biot-Savart Law
Biot-Savart Law was given by French Physicists Biot and Savart based on experiments they did.
Biot Savart's Law Diagram
Consider a finite conductor XY. This conductor XY carrying current I. Now, consider an infinitesimal element `dl ` of the conductor. Due to this element `dl` the magnetic field dB is to be determined at a point P which is at a distance r from it.
Let `\theta` be the angle between dl and the displacement vector r. The direction of vector `\vec {dl}` is in the direction of the current.
According to Biot-Savart Law
The magnitude of the magnetic field `\vec {dB}` is
☆ Proportional to the current I.
`|\vec {dB}| \prop I`
☆ Proportional to the element length `|\vec {dl}|`
`|\vec {dB}| \prop |\vec {dl}|`
☆ Proportional to the sine of the angle between vector `\vec r` and `\vec {dl}` that is `\theta`
`|\vec {dB}| \prop sin\theta`
☆ Inversely proportional to the square of the distance r.
`|\vec {dB}| \prop \frac{1}{r^2}`
Thus, in vector notation,
`|\vec {dB}| \prop \frac{I |\vec{dl}| sin\theta}{r^2}`
`|\vec {dB}| = \frac {\mu_0}{4 \pi} \frac{I |\vec {dl}| sin\theta}{r^2}`
`|\vec {dB}| = \frac {\mu_0}{4 \pi} \frac{I |\vec {dl}| \times \vec r }{r^3}`
or
`dB = \frac {\mu_0}{4 \pi} \frac{I dl sin\theta}{r^2}`
Where `\frac{\mu_0}{4 \pi}` is a constant of proportionality. The above expression is valid for vacuum.
Note: - `\vec {dl} \times \vec r` given by Right Hand Screw rule.
Value of `\frac{\mu_0}{4 \pi}`
☆ `\frac{\mu_0}{4 \pi} = 10^{-7} \frac {N}{m^2}`
Units of `\frac{\mu_0}{4 \pi}`
☆ ` \frac {N}{m^2}`
☆ `\frac {wb}{A\times m}`
☆ `\frac{T \times m}{A}`
Here, `\mu_0` is the permeability of free space (or vacuum).
For other medium
`|\vec {dB}| = \frac {\mu}{4 \pi} \frac{I |\vec {dl}| sin\theta}{r^2}`
Where, `\mu = \mu_0 \mu_r`, is the permeability of a specific medium, which is dependent on the medium.
Relative Permeability
`\mu_r = \frac{\mu_0}{ \mu_r}`
Biot-Savart Law in Vector Form
`\vec{dB} = \frac{\mu_0}{4 \pi} \frac{I \vec {dl} \times \hat r}{r^2}`
`\vec{dB} = \frac{\mu_0}{4 \pi} \frac{I \vec {dl} \times \vec r}{r^3}`
The direction of dB will be perpendicular to the surface formed by `\ved {dl}` and `\vec r` and would be up and down depending on the Right Handed Cork Screw Rule.
Right-Hand Screw Rule
The direction of `\vec{dl} \times \vec r` can be determined using the Right Hand Screw rule. According to this rule, if you rotate from the first vector `\vec {dl}` towards the second vector `\vec r` in an anticlockwise direction, the resulting vector `\vec{dl} \times \vec r` will point towards you. Conversely, if the rotation is clockwise, the resulting vector will point away from you. This rule helps to visualize the orientation of the magnetic field in relation to the current element and the displacement vector.
Similarities and Differences with Coulomg's Law
There are some similarities and differences between Biot-Savart's law for the magnetic field and Coulomb's Law.
Similiartities
Long Range: Both laws are long-range, and depend inversely on the square of the distance from the source to the point of interest. The principle of superposition applies to both fields.
Linearity : The magnetic field is linear in the source ` I dl `, just as the electrostatic field is linear in its source, the electric charge.
Differences
Scalar and Vector Source : The electrostatic field is produced by a scalar source, the electric charge, whereas the magnetic field is produced by a vector source `I \vec {dl}`.
Direction: The electrostatic field is along the displacement vector joining the source and the field point, while the magnetic field is perpendicular to the plane containing the displacement vector and the current element `I \vec {dl}`.
Angle Dependence: The Biot-Savart law includes an angle dependence not present in the electrostatic case. For Biot-Savart law if `\theta = 0^\circ` then `sin\theta = 0` and `dB = 0`, Thus the magnetic field at any point in the direction of `dl` is zero.
Relation Between `\epsilon_0`, `\mu_0`, and `c`:
Product of `\epsilon_0` and `\mu_0` is
`\epsilon_0 \mu_0 = 4 \pi \epsilon_0 \frac{\mu_0}{4 pi}`
`\epsilon_0 \mu_0 = \frac{1}{9 \times 10^ 9} \times 10^{- 7}`
`\epsilon_0 \mu_0 = \frac{1}{9 \times 10^ {9 + 7}`
`\epsilon_0 \mu_0 = \frac{1}{9 \times 10^ {16}`
`\epsilon_0 \mu_0 = \frac{1}{(3 \times 10^ 8)^2}`
`\epsilon_0 \mu_0 = \frac{1}{c^2}`
` c^2 = \frac{1}{\epsilon_0 \mu_0}`
` c = \frac{1}\sqrt{\epsilon_0 \mu_0}`
Where,
`\epsilon_0 =` The permittivity of free space or vacuum,
`\mu_0 =` The permeability of free space or vacuum,
`c =` Speed of light in vacuum
Thus, the speed of light in a vacuum is constant, and the product ` \epsilon_0 \mu_0` is fixed in magnitude. Choosing the value of either `\epsilon_0` or `\mu_0` fixes the value of the other. In SI units, `\mu_0` fixed to be equal to `4 \pi \times 10^{- 7} in magnitude.`
Applications of Biot-Savart's Law
☆ Biot-Savart's Law is used to calculate the magnetic field due to a current-carrying wire.
☆ Biot-Savart's Law is used to calculate the magnetic field due to a current-carrying solenoid.
☆ Biot-Savart's Law is used to calculate the magnetic field due to a current-carrying toroid.
NUMERICAL QUESTION
Q. Calculate the magnetic field `\vec {dB}` at a point P located 0.1 meters away form an infinitesimal element `dl` of length `1 \times 10^{-3}` meters carrying a current of 5 Amperes. The angle `\theta` between `dl` and the displacement vector `r` is `90^\circ`.
Q. A conductor segment of length `dl = 3 \times 10^{-3}` meters carries a current of a 20 Amperes. Calculate the magnetic field `\vec{dB}` at a distance r = 0.15 meters from the segment, assuming the angle `\theta` between `dl` and `r` is `60^\circ`.
QUESTIONS AND ANSWERS
Q. What is the value of the constant `frac{\mu_0}{4 \pi}` in the Biot-Savart Law?
Q. According to the Biot-Savart Law, on what factors does the magnitude of the magnetic field `\vec {dB}` depend?
Q. What is the relationship between the permeability of free space `\mu_0` and permeability of a specific medium `\mu`?
Q. How does the Biot-Savart Law for magnetic fields differ from Coulomb's Law for electrostatic fields in terms of their sources?
Q. Explain the angle dependence in Biot-Savart Law.
Q. Derive the relationship between the speed of light c, permittivity of free space `\epsilon_0`, and permeability of free space `\mu_0`.
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