\- In a cyclotron resonance experiment in potassium at \(68 \mathrm{GHz}\), three consecutive resonances were at magnetic fields of \(0.74 \mathrm{~T}, 0.59 \mathrm{~T}, 0.49 \mathrm{~T}\). Calculate the cyclotron effective mass of ele obse potassium.

Cyclotron resonance is a phenomenon observed when charged particles, such as electrons, move perpendicular to a uniform magnetic field. The particles spiral around the magnetic field lines. The cyclotron resonance condition is mathematically expressed using the formula: \[ \omega = \frac{e B}{m^*} \] Here, \(\omega\) is the angular frequency, \(e\) denotes the electron charge, \(B\) represents the magnetic field strength, and \(m^*\) is the cyclotron effective mass. This formula tells us how the magnetic field and the effective mass determine the resonance frequency.

In a cyclotron resonance experiment, the magnetic field strength \(B\) is crucial as it influences the resonance frequency of the particles. The magnetic field is measured in Tesla (T). For our exercise, we have three resonances occurring at magnetic fields of 0.74 T, 0.59 T, and 0.49 T.
This means the resonance condition is directly affected by these values. The higher the magnetic field, the higher the angular frequency for a given effective mass.

To calculate the cyclotron effective mass \(m^*\), we rearrange the cyclotron resonance formula. Using the relation: \[ m^* = \frac{e B}{\omega} \] we can insert known values for the electron charge \(e\), the magnetic field \(B\), and the angular frequency \(\omega\). For instance, given the frequency \(u = 68\) GHz, we convert it to angular frequency. Since \(\omega = 2\pi u\), we get: \[ \omega = 2\pi(68 \times 10^9) = 4.276 \times 10^{11} \text{ rad/s} \] Using \(e = 1.602 \times 10^{-19} \text{ C}\), and \(B = 0.74 \text{ T}\), we compute: \[ m^* = \frac{(1.602 \times 10^{-19} \text{ C}) (0.74 \text{ T})}{4.276 \times 10^{11} \text{ rad/s}} \approx 2.77 \times 10^{-31} \text{ kg} \] This process confirms that the cyclotron effective mass is critically influenced by the magnetic field and the frequency of the experiment.

Angular frequency \(\omega\) is a key concept in understanding cyclotron resonance. It is related to the regular frequency \(u\) by the formula: \[ \omega = 2 \pi u \] This converts the frequency measured in Hertz (Hz) to radians per second (rad/s). For example, if the experiment frequency is \(68\) GHz, then the angular frequency is calculated as: \[ \omega = 2\pi(68 \times 10^9) = 4.276 \times 10^{11} \text{ rad/s} \] Understanding this conversion is vital because it allows us to plug into the cyclotron resonance formula and accurately find the effective mass of the electron.