Question

Given below are two statements
Statement I: Biot-Savart's law gives on the expression for the magnetic field strength of an infinitesimal current element (Idl) of a current carrying conductor only.
Statement II: Biot-Savart’s law is analogous to Coulomb's inverse square law of charge q, with the former being related to the field produced by a scalar source, Idl while the latter being produced by a vector source, q.
In light of above statements choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct and Statement II is incorrect
• D. Statement I is incorrect and Statement II is correct

Answer 1

The Correct Option is C

Let's evaluate the given statements about Biot-Savart's law and determine their correctness:

Statement I: Biot-Savart's law gives an expression for the magnetic field strength due to an infinitesimal current element \( \mathbf{Idl} \) of a current-carrying conductor only.

This statement is correct. Biot-Savart's law describes how the magnetic field (\( \mathbf{B} \)) is generated by a small segment of a current-carrying wire. The magnetic field produced by an infinitesimal current element is given by:

\( d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat{r}}}{r^2} \)

where \( I \) is the current, \( d\mathbf{l} \) is the vector length of the infinitesimal segment, \( \mathbf{\hat{r}} \) is the unit vector from the element to the point where the field is measured, and \( r \) is the distance between the element and the point in question. This expression is specific to current elements.

Statement II: Biot-Savart's law is analogous to Coulomb's inverse square law of charge \( q \), with the former being related to the field produced by a scalar source, \( \mathbf{Idl} \), while the latter being produced by a vector source, \( q \).

This statement is incorrect. Biot-Savart's law and Coulomb's law are analogous in that they both involve inverse-square relationships for fields produced by charge or current elements. However, in Biot-Savart's law, the current element \( \mathbf{Idl} \) is a vector quantity, not a scalar. In contrast, the electric field in Coulomb’s law is produced by a scalar charge \( q \).

Conclusion: Considering the analysis above, the most appropriate answer is:

Statement I is correct and Statement II is incorrect

Answer 2

Step 1: Understand the Statements

We need to evaluate two statements:

Statement I: Biot-Savart's law gives the expression for the magnetic field strength of an infinitesimal current element (Idl) of a current-carrying conductor only.

Statement II: Biot-Savart's law is analogous to Coulomb's inverse square law of charge q, with the former being related to the field produced by a scalar source, q.

Let's analyze each statement to determine their correctness.

Step 2: Evaluate Statement I

Biot-Savart's law provides the magnetic field dB produced by a small current element Idl at a point in space. The formula is:

dB = (μ₀ / 4π) × (I dl × sinθ / r²)

where μ₀ is the permeability of free space, I is the current, dl is the length of the current element, r is the distance from the element to the point, and θ is the angle between the current element and the line connecting the element to the point.

The statement says this law applies "only" to an infinitesimal current element of a current-carrying conductor. This is correct because Biot-Savart's law is specifically formulated for an infinitesimal current element (Idl). To find the total magnetic field due to a larger conductor, we integrate over the entire length of the conductor. The "only" emphasizes that the law directly gives the field for a small element, not the entire conductor without integration.

So, Statement I is correct.

Step 3: Evaluate Statement II

Statement II claims that Biot-Savart's law is analogous to Coulomb's inverse square law, with the former being related to the field produced by a scalar source, q.

Let's break this down:

Part 1: Analogy to Coulomb's Law

Coulomb's law gives the electric field E due to a point charge q:

E = (1 / 4πε₀) × (q / r²)

where ε₀ is the permittivity of free space, q is the charge, and r is the distance from the charge. This is an inverse square law (1/r² dependence).

In Biot-Savart's law, the magnetic field dB also has a 1/r² dependence:

dB = (μ₀ / 4π) × (I dl × sinθ / r²)

Both laws have a similar mathematical form with an inverse square dependence on distance, and both involve fundamental constants (μ₀/4π for Biot-Savart and 1/4πε₀ for Coulomb). So, the analogy in terms of the mathematical structure (1/r²) holds true.

Part 2: Scalar Source, q

The second part of the statement says Biot-Savart's law is related to the field produced by a scalar source, q. This is problematic. In Coulomb's law, the source is a scalar charge q, producing an electric field. However, in Biot-Savart's law, the source of the magnetic field is a current element Idl, which is a vector quantity (since dl has direction). The magnetic field dB depends on the cross product I dl × r, which is inherently vectorial, not scalar. Magnetic fields are produced by moving charges (currents), not static scalar charges like q.

The statement incorrectly suggests that the source in Biot-Savart's law is a scalar quantity like q. In reality, the source is vectorial, and magnetic fields are not produced by scalar charges but by currents or moving charges.

So, while the analogy to Coulomb's law holds in terms of the inverse square dependence, the claim about the "scalar source, q" is incorrect. Therefore, Statement II is incorrect.

Step 4: Choose the Correct Option

Based on our analysis:

• Statement I is correct.
• Statement II is incorrect.

The correct option is: Statement I is correct and Statement II is incorrect, which corresponds to Option 3.

Final Answer: The correct answer is Option 3.