Radio astronomers detect electromagnetic radiation at 45 MHz from an interstellar gas cloud. They suspect this radiation is emitted by electrons spiraling in a magnetic field. What is the magnetic field strength inside the gas cloud?

Verified step by step guidance

1. Understand the problem: The radiation detected is due to electrons spiraling in a magnetic field, which is a phenomenon known as synchrotron radiation. The frequency of the radiation is related to the strength of the magnetic field and the motion of the electrons.
2. Use the formula for the cyclotron frequency (also called gyrofrequency), which relates the frequency of the emitted radiation to the magnetic field strength: $f=\frac{e}{2\pi m}B$, where $f$ is the frequency, $e$ is the charge of the electron, $m$ is the mass of the electron, and $B$ is the magnetic field strength.
3. Rearrange the formula to solve for the magnetic field strength $B$: $B=\frac{2\pi mf}{e}$.
4. Substitute the known values into the formula: the frequency $f$ is 45 MHz (convert this to Hz: $45\times 10^6$ Hz), the charge of the electron $e$ is approximately $1.6\times 10^-19$ C, and the mass of the electron $m$ is approximately $9.11\times 10^-31$ kg.
5. Perform the calculation to find the magnetic field strength $B$. Ensure all units are consistent (e.g., frequency in Hz, charge in Coulombs, mass in kilograms) and simplify the expression to determine the value of $B$.

Key Concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that travels through space at the speed of light. It encompasses a range of wavelengths, including radio waves, which are used in radio astronomy. The frequency of the radiation, measured in hertz (Hz), is directly related to its energy and can provide insights into the physical processes occurring in astronomical objects.

Cyclotron Radiation

Cyclotron radiation occurs when charged particles, such as electrons, spiral around magnetic field lines. This motion results in the emission of electromagnetic waves, with the frequency of the emitted radiation depending on the strength of the magnetic field and the charge-to-mass ratio of the particles. Understanding this phenomenon is crucial for interpreting the radiation detected from the gas cloud.

Magnetic Field Strength

Magnetic field strength, often denoted as B, is a measure of the intensity of a magnetic field at a given point in space. It influences the motion of charged particles and is a key factor in determining the frequency of cyclotron radiation. The relationship between the frequency of the emitted radiation and the magnetic field strength can be described by the formula f = (qB)/(2πm), where f is the frequency, q is the charge, B is the magnetic field strength, and m is the mass of the particle.