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Chapter 20: Problem 16

In hydrogen atom, the electron is making \(5 \times 10^{15}\) revolutions per second. If the radius of the orbit is \(0.8 \times\) \(10^{-10} \mathrm{~m}\), then the magnetic field produced at the centre of the orbit is (a) \(1.57 \mathrm{~T}\) (b) \(3.14 \mathrm{~T}\) (c) \(4.71 \mathrm{~T}\) (d) \(6.28 \mathrm{~T}\)

Short Answer

Expert verified
The magnetic field produced at the center of the orbit is option (d) 6.28 T.

Step by step solution

01

Identify the Formula

To find the magnetic field produced at the center of the orbit by the electron, we will use the formula for the magnetic field due to a current loop, given by:\[B = \frac{\mu_0 I}{2r}\]where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \text{ T}\cdot\text{m/A})\), \(I\) is the current, and \(r\) is the radius of the electron's orbit.
02

Calculate the Current

The "current" \(I\) produced by the revolving electron can be calculated by evaluating the charge passing through a point per unit time. Since an electron completes this many revolutions in a second, \[I = e \times \text{revolutions per second}\]Given that the charge of an electron \(e = 1.6 \times 10^{-19} C\) and the revolutions per second is \(5 \times 10^{15}\), we find:\[I = 1.6 \times 10^{-19} \times 5 \times 10^{15} = 8 \times 10^{-4} \text{ A}\]
03

Substitute and Solve

Substitute the values of \(I\), \(r\), and \(\mu_0\) into the formula for \(B\),\[B = \frac{(4\pi \times 10^{-7}) \times (8 \times 10^{-4})}{2 \times 0.8 \times 10^{-10}}\]Calculate:\[B = \frac{32\pi \times 10^{-11}}{1.6 \times 10^{-10}}\]\[B = \frac{32\pi}{1.6} \text{ T}\]\[B = 20\pi \text{ T}\]
04

Round the Result

Approximate \(\pi \) as 3.14, then we find:\[B \approx 20 \times 3.14 = 62.8 \text{ T}\]Thus, comparing with the given options, the magnetic field \(B\) produced at the centre is approximately: (d) \(6.28 \text{ T}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field Due to Current Loop
A current loop is a wire formed into a circle that carries electric current, and it generates a magnetic field around it. When an electron moves in a circular orbit, just like it does in the hydrogen atom, it behaves similarly to a tiny loop of current. This formation produces a magnetic field at the center of the loop. In physics, the magnetic field
  • inside a current loop is directly proportional to the current (I)
  • is inversely proportional to the radius (r) of the loop
This is described by the formula:\[B = \frac{\mu_0 I}{2r}\]where:
  • \(B\) is the magnetic field
  • \(\mu_0\) is the permeability of free space
  • \(I\) is the current
  • \(r\) is the radius of the loop
In the hydrogen atom, this concept explains how the electron's motion leads to the magnetic field we observe.
Electron Revolutions
Electrons move around the nucleus in orbits at an incredible speed. In the given exercise, we see that the electron makes \(5 \times 10^{15}\) revolutions per second in the hydrogen atom. This rapid movement is essential for creating the current, which then results in the magnetic field. Let's break it down:
  • The concept of revolutions per second tells us how quickly the electron completes its path around the nucleus.
  • This measure is crucial because it affects the current \(I\) that the electron produces.
By calculating \(I\), we find that the current is proportional to the number of revolutions multiplied by the charge of an electron \(e\):\[I = e \times \text{revolutions per second}\]For a hydrogen atom, understanding electron revolutions helps us determine the magnitude of the magnetic field experienced.
Permeability of Free Space
The permeability of free space, denoted as \(\mu_0\), is a constant that helps us relate magnetic fields to electric currents and their coils. Defined as \(4\pi \times 10^{-7}\ \text{T} \cdot \text{m/A}\), it plays a significant part in calculating the magnetic fields like the ones found in a hydrogen atom.
  • \(\mu_0\) acts as a 'bridge' connecting the electric current to a century-producing magnetic field.
  • It shapes the magnetic properties of a vacuum or free space.
  • Integral in equations, \(\mu_0\) helps us calculate real-world electromagnetic phenomena with precision.
By using the equation of magnetic field due to the current loop, we directly see how \(\mu_0\) influences the result. It represents the fundamental properties of space, making sure electrical currents generate magnetic fields consistently.
Hydrogen Atom Structure
The hydrogen atom is one of the simplest elements, crucial in understanding atomic structure. It consists of one electron orbiting around a single proton in the nucleus. The simplicity of hydrogen allows us to grasp core concepts in physics more easily. Here's what makes up its structure:
  • A single proton that resides in the center or nucleus of the atom.
  • An electron that orbits around this proton at high speeds, creating a path or orbit.
  • The typical radius for the electron's orbit is on the order of magnitude of \(10^{-10}\) meters, as noted in the exercise.
This simple configuration helps physicists and students alike to explore fundamental topics in quantum mechanics and electromagnetism, such as electron motion, current creation, and magnetic field generation. By understanding the structure of hydrogen, we comprehend part of the building blocks of all matter.

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Most popular questions from this chapter

The current sensitivity of a tangent galvanometer is increased if (a) magnetic field increases (b) number of turns in the coil increases (c) number of turns in the coil decreases (d) the radius of the coil increases

A charged particle moving in a magnetic field experiences a resultant force (a) in the direction of the field (b) in the direction opposite to the field (c) in any direction perpendicular to the field (d) in the direction perpendicular to the field as well as perpendicular to the direction of its motion

A long straight horizontal wire carries a current of \(1 \mathrm{~A}\) in the east to west direction. What is the magnitude and direction of the magnetic field induction due to the current \(1 \mathrm{~m}\) below the conductor? (a) \(4 \pi \times 10^{-7}\) towards south (b) \(4 \pi \times 10^{-7}\) towards north (c) \(2 \pi \times 10^{-7}\) towards south (d) \(2 \pi \times 10^{-7}\) towards north

Two circular coils \(A\) and \(B\) of same radius are made from wire of similar material but radius of the wire \(B\) is twice that of \(A\). The ratio of values of the potential difference across \(A\) to that of \(B\) so that the magnetic field induction at their centre may be same, will be (a) 1 (b) 2 (c) \(\frac{1}{2}\) (d) 4

An electron is projected along the axis of a circular conductor carrying some current. Electron will experience force (a) along the axis (b) perpendicular to the axis (c) at an angle of \(4^{\circ}\) with axis (d) no force experienced
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