Problem 1880 - JEE Physics

Question

0: Due to 10 Amp of current flowing in a circular coil of \(10 \mathrm{~cm}\) radius, the mag. field produced at its centre is \(\pi \times 10^{-3}\) Tesla. The number of turns in the coil will be (a) 5000 (b) 100 (c) 50 (d) 25

Step-by-Step Solution

Answer

The number of turns in the coil is 50 (option c).

Step 1: Write down the given information and the formula

Given: Current, I = 10 A Radius, r = 10 cm = 0.1 m (converted to meters) Magnetic field at the center, B = \(\pi \times 10^{-3}\) T The formula for the magnetic field produced by a current-carrying circular coil at its center is: \[B = \frac{\mu_0 n I}{2r}\] where B is the magnetic field, \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} \mathrm{Tm/A}\)), n is the number of turns, I is the current, and r is the radius of the coil.

Step 2: Solve for the number of turns (n)

We need to find the value of n. So, we will rearrange the formula for n and then substitute the given values. \[n = \frac{2rB}{\mu_0 I}\] Substituting the given values: \[n = \frac{2(0.1)(\pi \times 10^{-3})}{(4\pi \times 10^{-7})(10)}\]

Step 3: Calculate the number of turns

Now, we will perform the calculation: \[n = \frac{(0.2\pi \times 10^{-3})}{(4\pi \times 10^{-7})(10)}\] \[n = \frac{0.2\pi \times 10^{-3}}{4\pi \times 10^{-6}}\] \[n = \frac{0.2}{4} \times 10^{3}\] \[n = 0.05 \times 10^{3}\] \[n = 50\] So, the number of turns in the coil is 50. The correct option is (c) 50.

Key Concepts

Circular Coil

Imagine a circular loop or coil of wire, which might remind you of a slinky toy but in a flat shape. This coil is crucial in magnetic field exercises, particularly in understanding how electricity interacts with magnetism.
The coil can carry electrical current. The interesting part is that when electricity flows through it, a magnetic field is generated around the coil. This is why coils are often utilized in devices like transformers and electric motors.
The circular shape is significant because it allows the magnetic field lines to focus at the center of the coil. This creates a stronger magnetic field at that point compared to other shapes. Knowing this, scientists and engineers often calculate how changes in the coil's properties—like size or material—affect the magnetic field strength, which is measured in Tesla.

• This knowledge helps in designing efficient electronic devices.
• It's a practical application of Ampère's circuital law and electromagnetic induction.

Current

Current is the flow of electric charge. You can think of it as a river of electricity flowing through wires and components. In physics, it's measured in Amperes (A). In the problem, the coil carries a current of 10 Amperes.
The amount of current directly influences how strong the magnetic field will be.
A higher current leads to a stronger magnetic field when flowing through a circular coil.

• Electrical current's flow is guided by voltage (or electrical potential).
• Ohm's Law explains the relationship: Current = Voltage/Resistance.

Hence, controlling the amount of current is crucial in shaping the magnetic field generated by a coil, making it more efficient for specific tasks, like in electromagnets or communication devices.

Number of Turns

The number of turns in a coil refers to how many loops or circles the wire makes in the coil. More turns mean the wire wraps around more times. This count is important because it affects the magnetic field's strength.
In our exercise, the coil's magnetic field strength at the center depends on the number of loops or turns it has.

• More turns result in a stronger magnetic field.
• The relationship is direct: Double the turns, double the field strength (assuming other factors remain constant).

In our example, we calculated the number of turns needed in the coil to be 50 using the given formula for magnetic field strength. This formula integrates all coil properties into a single factor, helping anticipate how the coil will perform in various settings.

Permeability of Free Space

This might sound a bit complex, but think of it as a property of space that affects how magnetic fields form and spread. The permeability of free space, often represented as \(\mu_0\), is a constant value that represents the extent to which space allows magnetic fields to pass through. This factor is critical when calculating magnetic fields in coil problems.
In the context of our problem, \(\mu_0\) has a constant value of \(4\pi \times 10^{-7} \ \mathrm{Tm/A}\).
This value helps determine how effectively a magnetic field can be created in a vacuum and is used in the formula to calculate magnetic field strength in magnetic exercises involving coils.

• A lesser known fact is that this constant forms part of the Maxwell equations, describing how electric and magnetic fields interact.

Understanding this constant helps in calculating accurate magnetic fields in laboratory and industrial settings, ensuring that designs and experiments involving magnetism are very precise.