Question

In each of the following questions, Match column-I and column-II and select the correct match out of the four given choices.$$\text{Unknown environment 'tabular'}$$ (A) Biot-savart's law (P) Direction of magnetic field induction (B) Right hand thumb rule (Q) Magnitude of magnetic field induction (C) Fleming's left hand rule (R) Direction of induced current (D) Fleming's right hand rule (S) Direction of force due to a mag. field (a) $\mathrm{A}\to \mathrm{Q};\mathrm{B}\to \mathrm{P};\mathrm{C}\to \mathrm{R};\mathrm{D}\to \mathrm{S}$ (b) $\mathrm{A}\to \mathrm{Q};\mathrm{B}\to \mathrm{P};\mathrm{C}\to \mathrm{S};\mathrm{D}\to \mathrm{R}$ (c) $\mathrm{A}\to \mathrm{P};\mathrm{B}\to \mathrm{Q};\mathrm{C}\to \mathrm{R};\mathrm{D}\to \mathrm{S}$ (d) $\mathrm{A}\to \mathrm{P}:\mathrm{B}\to \mathrm{Q}:\mathrm{C}\to \mathrm{S};\mathrm{D}\to \mathrm{R}$

Short answer

The correct match for the given columns is (b) $\mathrm{A}\to \mathrm{Q};\mathrm{B}\to \mathrm{P};\mathrm{C}\to \mathrm{S};\mathrm{D}\to \mathrm{R}$.

Step-by-step solution

Identifying the principles and their properties

(A) Biot-Savart's Law is used to find the magnetic field induced by a current-carrying conductor. It is mainly used for the magnitude and direction of the magnetic field. → The correct match for this is Magnitude of magnetic field induction (Q). (B) The Right Hand Thumb Rule (also known as the Right-Hand Grip Rule) is used to find the direction of the magnetic field due to a current-carrying conductor (wire). → The correct match for this is Direction of magnetic field induction (P). (C) Fleming's Left Hand Rule is used to find the direction of force experienced by a current-carrying wire in a magnetic field. → Correct match is Direction of force due to a magnetic field (S). (D) Fleming's Right Hand Rule is used to determine the direction of induced current in a conductor moving in a magnetic field. → Correct match is Direction of induced current (R).

Selecting the correct match from the given options

Now, we will compare the connections we have made in Step 1 with the given options: (a) A → Q; B → P; C → R; D → S (b) A → Q; B → P; C → S; D → R (c) A → P; B → Q; C → R; D → S (d) A → P; B → Q; C → S; D → R We found the connections to be A → Q; B → P; C → S; D → R which matches with the option (b). Hence, the correct answer is (b) $\mathrm{A}\to \mathrm{Q};\mathrm{B}\to \mathrm{P};\mathrm{C}\to \mathrm{S};\mathrm{D}\to \mathrm{R}$.

Key concepts

Biot-Savart Law

The Biot-Savart Law is essential for understanding magnetic fields. It helps us calculate the magnitude and direction of the magnetic field created by a small segment of a current-carrying wire. This law is particularly useful when dealing with simple shapes, like a loop or a straight line, to determine how electric currents generate magnetic fields.
According to the Biot-Savart Law, the magnetic field ($B$) at a point in space is directly proportional to the current ($I$), the length of the wire ($dL$), and the sine of the angle ($\theta$) between the wire and the point. In mathematical terms, this is expressed as:
$$dB=\frac{{\mu }_0}{4\pi }\cdot \frac{I\cdot dL\cdot \sin \theta }{r^2}$$
where ${\mu }_0$ is the permeability of free space, and $r$ is the distance from the wire to the point of interest.
The Biot-Savart Law is fundamental in electromagnetic theory and is key to understanding how currents can shape magnetic fields. It's often the first step toward deeper concepts like Ampère's Law or when dealing with complex current paths.

Right Hand Thumb Rule

The Right Hand Thumb Rule is a simple tool for finding the direction of a magnetic field around a straight conductor. It is sometimes called the Right-Hand Grip Rule. This rule states that if you wrap your right hand around the wire with your thumb pointing in the direction of the current, your fingers will naturally curl in the direction of the magnetic field lines.
This visualization helps not only students but also professionals to quickly determine field directions without complex calculations. The mnemonic is straightforward:

• Thumb: Direction of current.
• Fingers: Direction of the magnetic field loop.

This rule is particularly useful when creating or interpreting circuit diagrams, giving a practical understanding of how electric currents generate magnetic fields. It places an emphasis on the interdependent relationship between electricity and magnetism.

Fleming's Left Hand Rule

Fleming's Left Hand Rule is a visualization technique used to determine the direction of force acting on a conductor within a magnetic field. This rule is fundamental in applications such as electric motors.
According to this rule, you arrange your fingers of the left hand at right angles to each other with:

• Thumb: representing the Motion or force.
• Forefinger: representing the magnetic Field (from north to south).
• Middle finger: representing the Current (from positive to negative).

When the current flows through the wire within a magnetic field, it experiences a force, making this rule indispensable for understanding motor principle and applying it to real-life devices. It gives insight into mechanical actions derived from electromagnetic forces.

Fleming's Right Hand Rule

Fleming’s Right Hand Rule aids in finding the direction of induced current when a conductor moves through a magnetic field. This principle is pivotal for understanding how generators work.
Set up your right hand with your fingers perpendicular:

• Thumb: Motion of the conductor.
• Forefinger: Magnetic Field (north to south direction).
• Middle finger: Induced Current.

In generators, as a coil turns through the magnetic field, an electromotive force (EMF) is induced, and this rule helps predict the current direction. Being able to visualize these components is crucial in designing and interpreting the workings of electrical generators, ensuring energy conversion processes are understood effectively.
Fleming's Right Hand Rule bridges the connection between mechanical movement and electrical energy, giving insights into the fundamental principles of electromagnetism.