Problem 1902 - JEE Physics

Question

A straight wire of length \(30 \mathrm{~cm}\) and mass 60 milligram lies in a direction \(30^{\circ}\) east of north. The earth's magnetic field at this site is horizontal and has a magnitude of \(0.8 \mathrm{G}\). What current must be passed through the wire so that it may float in air ? \(\left[\mathrm{g}=10\left(\mathrm{~m} / \mathrm{s}^{2}\right)\right]\) (a) \(10 \mathrm{Amp}\) (b) \(20 \mathrm{Amp}\) (c) \(40 \mathrm{Amp}\) (d) \(50 \mathrm{Amp}\)

Step-by-Step Solution

Answer

The required current for the wire to float in air is approximately \(50 A\), which corresponds to option (d).

Step 1: Calculate the weight of the wire

First, we need to find the gravitational force acting on the wire. The weight of the wire can be determined using the formula: Weight (W) = mass (m) × gravitational acceleration (g) Given, mass of the wire (m) = 60 milligram = 60 × 10^(-3) kg (converting to kg) Gravitational acceleration (g) = 10 m/s² Weight (W) = 60 × 10^(-3) kg × 10 m/s² = 0.6 N

Step 2: Set up the equilibrium condition

The wire floats in the air due to the balance between the magnetic force (F_m) and the weight of the wire (W). The magnetic force on the wire can be calculated using this formula: F_m = B × I × L × sin(θ) Where: - B is the magnetic field (0.8 G = 0.8 × 10^(-4) T, converting to Tesla) - I is the current that we need to find - L is the length of the wire (30 cm = 0.3 m, converting to meters) - θ is the angle between the wire and the direction of the magnetic field (30 degrees) Since the wire is floating, the magnetic force (F_m) is equal to the weight (W), so: F_m = W

Step 3: Solve for the current (I)

Using the equilibrium condition (F_m = W), we can solve for the current (I): 0.8 × 10^(-4) T × I × 0.3 m × sin(30°) = 0.6 N Dividing both sides by (0.8 × 10^(-4) T × 0.3 m × sin(30°)), we get: I = 0.6 N / (0.8 × 10^(-4) T × 0.3 m × sin(30°)) I = 0.6 N / (0.00008 T × 0.3 m × 0.5) I ≈ 50 A So, the required current for the wire to float in air is approximately 50 A, which corresponds to option (d).

Key Concepts

Current in a Wire

Current in a wire is the flow of electric charge along the length of the wire. The movement is usually due to electrons moving through a conductor, like a metal wire.
To have a constant flow of current, a potential difference (voltage) must be applied across the wire. This causes the free electrons in the wire to flow from a region of high potential to a region of low potential.

• The direction of current flow is conventionally taken as opposite to the direction of electron flow.
• Current is measured in Amperes (A), which indicates the number of charges passing a point per unit time.

In the scenario of a wire in a magnetic field, the flow of current interacts with the magnetic field, resulting in a force acting on the wire. This force will depend on several factors, including the amount of current flowing through the wire and the strength of the magnetic field.

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. Magnetic fields are represented by field lines that go from the north pole to the south pole of a magnet.
In physics, the unit of the magnetic field is the Tesla (T). However, smaller fields like the Earth's magnetic field are often measured in Gauss (G), where 1 G = 10^(-4) T.

• Magnetic fields exerts a force on ions and other moving electric charges in the field.
• The direction of the magnetic force on a moving charge is perpendicular both to the velocity of the charge and to the magnetic field itself, following the right-hand rule.

When a straight wire carrying a current lies in a magnetic field, the magnetic force can alter the wire's motion or position based on the interaction dynamics described by Lorentz force law.

Equilibrium Condition

An equilibrium condition is achieved when two forces acting on a body are equal in magnitude and opposite in direction, resulting in no net force and therefore no change in the state of motion of the body. This can be applied to static objects or bodies in uniform motion.
For the wire to "float" in air, it must reach an equilibrium condition where the magnetic force acting upwards on the wire is exactly balanced by the gravitational force (weight) acting downwards.

• This gravitational force is determined by the product of the mass of the wire and the gravitational acceleration.
• Meanwhile, the magnetic force is influenced by the current, the magnetic field, the length of the wire, and the angle between the wire and the magnetic field.

Equilibrium is achieved at the specific current value that satisfies the balance of these forces, resulting in a stable position for the wire.