Problem 53 - Master Resource Book in JEE Main Physics

Question

Which of the following relations represent Biot-Savart's law ? (a) \(\mathrm{dB}=\frac{\mu_{0}}{4 \pi} \frac{i d l}{r} \hat{\mathbf{r}}\) (b) \(\mathrm{dB}=\frac{\mu_{0}}{4 \pi} \frac{i d l}{r^{2}} \mathrm{r}\) (c) \(\mathrm{dB}=\frac{\mu_{0}}{4 \pi} \frac{i \mathrm{~d} \mathrm{l} \times \mathrm{r}}{r^{3}}\) (d) \(\mathrm{dB}=\frac{\mu_{0}}{4 \pi} \frac{i \mathrm{dl}}{r^{4}} \hat{\mathrm{r}}\)

Step-by-Step Solution

Answer

Option (c) represents Biot-Savart's law: A ext{dB} = rac{16_0}{4 316} rac{i ext{d} ext{l} imes ext{r}}{r^3}A.

Step 1: Understanding Biot-Savart Law

The Biot-Savart law relates to magnetic fields generated by currents. It states that the differential magnetic field 8 ext{dB}9 at a point in space is proportional to the current 8i9 and the differential length element 8 ext{d} ext{l}9, and inversely proportional to the square of the distance 8r^29. The direction of 8 ext{dB}9 is given by the right-hand rule involving the cross product of 8 ext{d} ext{l}9 and 8 ext{r}9, resulting in the formula: A ext{dB} = rac{16_0}{4 316} rac{i ext{d} ext{l} imes ext{r}}{r^3}A.

Step 2: Identify the Correct Formula

Biot-Savart law involves a cross product of the differential length element 8 ext{d} ext{l}9 and the position vector 8 ext{r}9 divided by the cube of the distance (A r^3A). We are looking for the answer that matches the form: A ext{dB} = rac{16_0}{4 316} rac{i ext{d} ext{l} imes ext{r}}{r^3}A.

Step 3: Matching Choice

Examine the provided options: (a), (b), (c), and (d). Option (c) is A ext{dB} = rac{16_0}{4 316} rac{i ext{d} ext{l} imes ext{r}}{r^3}A, which precisely matches the form of Biot-Savart's law as it includes the cross product and division by 8r^39.

Key Concepts

Magnetic Fields

Magnetic fields are invisible forces that affect the motion of charged particles like electrons. When you bring a magnet close to a piece of metal, you're observing the effects of a magnetic field. In the context of electromagnetism, these fields are crucial because they exert a force not only on ferromagnetic materials but also on moving charges.

Magnetic fields can be visualized as lines that emerge from the north pole of a magnet and loop back into the south pole. These field lines never cross one another and the closer they are, the stronger the magnetic field at that point.

Given their impact, magnetic fields have a wide array of applications, from data storage devices to MRI machines in hospitals, demonstrating their undeniable role in modern technology.

Current and Magnetism

In physics, current refers to the flow of electric charge. When this flow occurs in a conductor, such as a wire, it generates a magnetic field around the conductor. This connection between current and magnetism allows us to create electromagnets, which are magnets powered by electricity.

When current flows through a coil of wire, the magnetic field is concentrated inside the coil and behaves similarly to a bar magnet. The strength of this electromagnetic field depends on several factors, including the amount of current, the number of loops in the coil, and the core material around which the coil is wound.

Understanding the relationship between current and magnetism is essential in electrical engineering and physics, aiding in the design of motors, transformers, and generators.

Electromagnetism

Electromagnetism is one of the fundamental forces of nature and is responsible for almost all the phenomena encountered in daily life, except gravity. It holds electrons in atoms and is the force behind electricity, magnetism, and light. James Clerk Maxwell's equations describe how electric charges and currents create electric and magnetic fields, and how they interact.

The sector of electromagnetism that deals with the interaction of electric currents and magnetic fields includes the Biot-Savart Law. This law helps you calculate the magnetic field generated by a small segment of current-carrying wire. It's very similar to how Coulomb's Law describes the electric field created by static charges.

Through the study of electromagnetism, we can better understand and harness electricity and magnetism to improve technology, ranging from power grids to electronics.

Right-Hand Rule

The right-hand rule is a simple way to determine the direction of a magnetic field or force if the direction of current is known. It's a handy mnemonic used in physics and engineering, particularly in electromagnetism, to figure out vector directions in three-dimensional space.

When using the right-hand rule for a current-carrying wire, point your thumb in the direction of current flow. Your curled fingers will mimic the direction of the magnetic field lines around the wire.

In the context of Biot-Savart's Law, the right-hand rule helps establish the direction of the magnetic field created by the current element. It's an essential tool for visualizing and solving problems involving cross products. Whether determining forces in a motor or understanding how fields interact, the right-hand rule makes those invisible forces more tangible.