Problem 1889 - JEE Physics

Question

For the mag. field to be maximum due to a small element of current carrying conductor at a point, the angle between the element and the line joining the element to the given point must be (a) \(0^{\circ}\) (b) \(90^{\circ}\) (c) \(180^{\circ}\) (d) \(45^{\circ}\)

Step-by-Step Solution

Answer

For the magnetic field to be maximum due to a small element of current-carrying conductor at a point, the angle between the element and the line joining the element to the given point must be \(90^{\circ}\), according to the Biot-Savart law. Therefore, the correct answer is (b) \(90^{\circ}\).

Step 1: Understand Biot-Savart law

The Biot-Savart law helps us find the magnetic field (dB) produced by a small current element (Idl). It states that the magnitude of the magnetic field produced is directly proportional to the current (I), the length of the small current element (dl), the sine of the angle (θ) between the line joining the element and the given point, and inversely proportional to the square of the distance (r) from the element to the given point. The formula for Biot-Savart law is given by: \( dB = \frac{\mu_0}{4\pi} \frac{I(dl \times r \sin(\theta))}{r^2} \) where µ₀ is the permeability of free space.

Step 2: Analyzing the formula for maximum magnetic field

In order to maximize dB, we need to find the maximum value of \(\sin(\theta)\). Since the sine function has a maximum value of 1, this occurs when the angle θ is equal to \(90^{\circ}\). The formula then simplifies to: \( dB = \frac{\mu_0}{4\pi} \frac{I(dl \times r)}{r^2} \)

Step 3: Identifying the correct angle

As we see from the analysis in Step 2, the maximum magnetic field occurs when \(\theta = 90^{\circ}\). Therefore, the answer is (b) \(90^{\circ}\).

Key Concepts

Magnetic Field

In physics, the magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetic materials. It is denoted by the letter \( \mathbf{B} \) and can exert a force on moving charges and materials intrinsically sensitive to magnetism. This force is often observed around permanent magnets or electric currents. When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field. The strength and direction of a magnetic field can vary based on the configuration of the sources creating the field.

• The Biot-Savart Law is an essential tool to determine the magnetic field at a specific point due to a current-carrying conductor.
• The size of the magnetic field depends on several factors, including the magnitude of the current, the specific geometry of the current path, and the distance from the current to the point of interest.
• Magnetic fields are crucial in technology, especially in electric motors, generators, and magnetic storage media.

Current Element

A current element is a small segment of an electrical current-carrying conductor. It's an essential concept in understanding how magnetic fields are generated by current in a material. Each current element contributes to the overall magnetic field produced by the conductor. In the Biot-Savart law, a current element is represented as \( I \, dl \), where \( I \) is the current through the conductor and \( dl \) is the differential length element of the conductor.

• The current element is often considered infinitesimally small to simplify calculations, especially when using integral calculus to determine the field of a conductor that isn’t straight or uniform.
• Each of these small elements creates its own magnetic field that combines with others to form the overall magnetic field surrounding the conductor.
• This principle is essential in the design of electric circuits and devices such as electromagnets and transformers.

Sine Function

The sine function is a mathematical function found in trigonometry that describes a smooth oscillation. In the context of Biot-Savart Law, the sine function helps to calculate the component of the current element's magnetic field that contributes to the field at a certain point in space. The angle \( \theta \) in the sine function is the angle between the current element and the line joining the element to the point where the magnetic field is being measured.

• The value of \( \sin(\theta) \) varies between -1 and 1.
• The magnetic effect is maximal when \( \theta = 90^\circ \) (\( \sin(90^\circ) = 1 \)), as this means the full magnetic influence of the current element is being measured at that point.
• Using \,\( \sin(0^\circ) = 0 \) and \,\( \sin(180^\circ) = 0 \), no magnetic field is produced in these angular positions.

Understanding the nature of the sine function in this relationship is crucial for optimizing and predicting magnetic field strengths in practical applications.

Angle Optimization

Angle optimization is a crucial aspect of maximizing the magnetic field strength at a certain point due to a current element. In solving magnetic field problems using the Biot-Savart Law, finding the optimal angle that results in the strongest magnetic field is essential. The sine function component of the Biot-Savart formula reveals that the magnetic field is maximized when the angle \( \theta = 90^\circ \). This happens because the value of \( \sin(\theta) \) reaches its maximum of 1 at this angle.

Some key points about angle optimization include:

• Geometrically, a \( 90^\circ \) angle means that the current element and the point, where the field is measured, are perpendicular to each other.
• Understanding this principle is vital for designing effective and efficient magnetic systems, such as those in wireless charging or efficient power transmission systems.
• This concept ensures optimized performance in electronic devices that rely on magnetic fields for their operation, reducing power waste and improving overall field efficiency.

Mastering angle optimization helps engineers and physicists design better electromagnetic systems tailored to meet specific needs.