Question

5: When the current flowing in a circular coil is doubled and the number of turns of the coil in it is halved, the magnetic field at its centre will become (a) Four times (b) Same (c) Half (d) Double

Short answer

The magnetic field at the center of the circular coil will become double when the current is doubled and the number of turns is halved ($B_2=2B_1$).

Step-by-step solution

Formula for Magnetic Field at the Center of a Circular Coil

The formula for the magnetic field (B) at the center of a circular coil with radius R, carrying a current I, and having n turns is given by: $B=\frac{{\mu }_0}{4\pi }\frac{2\pi nI}{R}$ Where ${\mu }_0$ is the permeability of free space. 2. Calculate the initial magnetic field

Initial Magnetic Field

Let's denote the initial magnetic field as $B_1$. Using the formula, we have: $B_1=\frac{{\mu }_0}{4\pi }\frac{2\pi nI}{R}$ 3. Apply the given changes to the coil, current, and number of turns

Changes in Current and Number of Turns

The problem states that the current flowing in the coil is doubled, and the number of turns of the coil is halved. So, we can write the new current as $2I$, and the new number of turns as $\frac{n}{2}$. 4. Calculate the new magnetic field

New Magnetic Field

Let's denote the new magnetic field as $B_2$. Using the formula with the new current and number of turns, we have: $B_2=\frac{{\mu }_0}{4\pi }\frac{2\pi (\frac{n}{2})(2I)}{R}$ 5. Compare the initial and new magnetic field

Comparison of Magnetic Fields

To find the relationship between the initial and new magnetic field, let's divide $B_2$ by $B_1$: $\frac{B_2}{B_1}=\frac{\frac{{\mu }_0}{4\pi }\frac{2\pi (\frac{n}{2})(2I)}{R}}{\frac{{\mu }_0}{4\pi }\frac{2\pi nI}{R}}=\frac{(\frac{n}{2})(2I)}{nI}=2$ 6. Select the correct option

Correct Option

We found that the new magnetic field is two times the initial magnetic field ($B_2=2B_1$). Therefore, the correct answer is: (d) Double

Key concepts

Electromagnetism

Electromagnetism is a branch of physics that focuses on the study of electromagnetic forces. These forces arise due to the interaction between electrically charged particles. An essential aspect of electromagnetism is the way electric currents and magnetic fields relate to one another. When an electric current flows through a wire, it generates a magnetic field around it. This principle is vital when understanding applications like motors, transformers, and coils.

In the context of a circular coil, electromagnetism explains how the movement of electrons (current) through the coil produces a magnetic field at its center. The strength and direction of this magnetic field are influenced by factors such as the amount of current, the number of turns in the coil, and the coil's radius. The behavior of these elements is predictable by using specific formulas, helping us solve particular physics problems. By comprehending electromagnetism, students grasp the foundational theories driving many modern technologies.

Ampere's Law

Ampere's Law is a crucial equation in electromagnetism that links electric currents with magnetic fields. Named after the French physicist André-Marie Ampère, the law states that the integrated magnetic field around a closed loop is proportional to the electric current passing through it.

This can be mathematically expressed as:$$\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_{0}I$$where:

• $\mathbf{B}$ is the magnetic field,
• $d\mathbf{l}$ is a differential element of the loop,
• ${\mu }_0$ is the permeability of free space,
• $I$ is the net current through the loop.

In practical terms, Ampere's Law helps in calculating the strength of the magnetic field in different parts of a circuit. For a circular coil, we see how the current and turns affect the magnetic field using these principles. The law enriches our understanding by simplifying the often complex interactions between currents and magnetic fields.

Physics Problem Solving

Physics problem solving involves applying mathematical formulas and theories to understand and predict physical phenomena. It requires a combination of logical reasoning, theoretical knowledge, and the ability to interpret equations.

When solving a physics problem, like determining the magnetic field of a circular coil, follow a strategic approach:

• Identify what is being asked and the relevant physical principles.
• Write down known information and variables, such as current and number of turns.
• Apply the appropriate formula, in this case, the one that links magnetic fields to currents and number of turns.
• Perform calculations methodically to solve for the unknown quantity.
• Analyze the results to ensure they make sense physically.

In this specific exercise, we saw how changes to current and coil turns affected the magnetic field. The adjustments indicate why understanding formulas and solving step-by-step is essential. By honing these skills, students can tackle a wide range of physics problems efficiently and accurately.