Problem 1883 - JEE Physics

Question

In a H-atom, an electron moves in a circular orbit of radius \(5.2 \times 10^{-11}\) meter and produces a mag. field of \(12.56\) Tesla at its nucleus. The current produced by the motion of the electron will be (a) \(6.53 \times 10^{-3}\) (b) \(13.25 \times 10^{-10}\) (c) \(9.6 \times 10^{6}\) (d) \(1.04 \times 10^{-3}\)

Step-by-Step Solution

Answer

The current produced by the motion of the electron in the given hydrogen atom is approximately \(1.04 \times 10^{-3} A\). The correct answer is (d) \(1.04 \times 10^{-3}\).

Step 1: Write down the relevant formula for magnetic field produced by a current-carrying loop

The magnetic field produced by a current-carrying loop at its center is given by the formula: \(B = \dfrac{\mu_0 I}{2 r}\), where - \(B\) is the magnetic field, - \(\mu_0\) is the magnetic constant (\(4\pi \times 10^{-7} Tm/A\)) - \(I\) is the current, - and \(r\) is the radius of the loop.

Step 2: Solve for the current (I) using the given values

We are given the magnetic field \(B = 12.56\) T and the radius \(r = 5.2 \times 10^{-11}\) m. Plug these values into the formula and solve for the current: \(I = \dfrac{2 r B}{\mu_0}\). Substituting the values, we have: \(I = \dfrac{2 \times (5.2 \times 10^{-11}) \times 12.56}{(4\pi \times 10^{-7})}\)

Step 3: Calculate the current I

Calculate the current I: \(I = \dfrac{2 \times (5.2 \times 10^{-11}) \times 12.56}{(4\pi \times 10^{-7})} \approx 1.04 \times 10^{-3} A\) Thus, the current produced by the motion of the electron is approximately \(1.04 \times 10^{-3} A\). The correct answer is (d) \(1.04 \times 10^{-3}\).

Key Concepts

Magnetic Field

A magnetic field is an invisible force field that surrounds magnetic objects and moving charges. It arises due to the movement of electric charge, such as an electron in orbit. In the context of atoms, the motion of an electron in its circular path creates a magnetic field similar to a tiny loop of current.

• The strength of a magnetic field is typically measured in teslas (T), named after the physicist Nikola Tesla.
• Magnetic fields can influence other magnetic objects and charges within their vicinity, resulting in forces that act on them.

Additionally, magnetic fields play a crucial role in atoms. They can influence how atoms interact with external magnetic fields, leading to phenomena like Zeeman splitting where spectral lines are affected by the presence of a magnetic field.

Current in Circular Motion

When an electron moves in a circular path, it produces a current similar to how a circular loop with flowing electrons forms a current. This is a significant concept as it connects the motion of subatomic particles to the larger contexts of electromagnetism.

• A current is a flow of electric charge, and when charges move in a circle, they produce effects similar to a loop with consistent current flow.
• The direction of the current is defined by the path's orientation, which can be analyzed using the right-hand rule.

Understanding this helps explain why atoms have magnetic properties, as the circular motion of electrons generates currents, thereby inducing magnetic fields at their nuclei.

Hydrogen Atom

The hydrogen atom is the simplest atom, composed of just one proton and one electron. This simplicity makes it an excellent model for exploring fundamental concepts in physics. The electron orbits the nucleus in specific paths known as orbitals.

• In classical physics, the electron is considered to be orbiting the nucleus in a circle with a specified radius.
• The calculations for properties like magnetic fields involve considering the magnetic moment produced by this motion.

This understanding helps bridge classical and quantum physics, as the electron's orbit in the hydrogen atom can be analyzed to reveal both magnetic and electrical characteristics.

Physics Formulas

Physics formulas enable us to calculate the different properties and behaviors of systems like atoms and electrons. These formulas are grounded in laws discovered through observation and experimentation. For instance, the formula used to calculate the magnetic field tells us how strong the field is based on current and distance.

• In this exercise, the magnetic field formula, \(B = \dfrac{\mu_0 I}{2 r}\), relates the magnetic field \(B\) to the current \(I\) and radius \(r\).
• Understanding these relationships helps in predicting and explaining physical phenomena.

Using such formulas ensures that complex physical interactions can be quantified and understood, allowing scientists and engineers to develop technologies and theoretical models effectively.