Question

The operating equation for a synchrotron in the relativistic range is $$ q B=\omega m\left[1-(\omega R)^{2} / c^{2}\right]^{-1 / 2} $$ where \(q\) and \(m\) are the charge and rest mass of the particle being accelerated, \(B\) is the magnetic field strength, \(R\) is the orbit radius, \(\omega\) is the angular frequency, and \(c\) is the speed of light. If \(\omega\) and \(B\) are varied (all other quantities constant), show that the relation between \(d \omega\) and \(d B\) can be written as $$ \frac{d B}{B^{3}}=\left(\frac{q}{m}\right)^{2} \frac{d \omega}{\omega^{3}} $$ or as \(\quad \frac{d B}{B}=\frac{d \omega}{\omega}\left[1-(\omega R / c)^{2}\right]^{-1}.\)

Short answer

\( \frac{dB}{B^{3}} = \left(\frac{q}{m}\right)^{2} \frac{d\omega}{\omega^{3}} \sim \frac{dB}{B} = \frac{d\omega}{\omega}\left[1 - (\omega R / c)^{2}\right]^{-1} \).

Step-by-step solution

Differentiate the given equation

Start by differentiating both sides of the equation \(q B = \omega m\left[1-(\omega R)^{2} / c^{2}\right]^{-1 / 2}\) with respect to \(\omega\).

Apply the product rule

Since \(q\) and \(m\) are constants, apply the product rule on the left-hand side and the chain rule on the right-hand side. The differentiation gives us:\(q \frac{dB}{d\omega} = m \left(-\frac{1}{2} \right) \left(1-(\omega R)^{2}/c^{2}\right)^{-3/2} \cdot \left(-2 \omega R^2 / c^2 \right)\).

Simplify the differentiated equation

Simplify the expression:\[q \frac{dB}{d\omega} = m \left[1-(\omega R)^{2} / c^{2}\right]^{-3/2} \cdot \left(\omega R^{2} / c^{2}\right).\]

Substitute back the original equation

Now, recall the original equation \(q B = \omega m\left[1-(\omega R)^{2} / c^{2}\right]^{-1 / 2}\) and isolate the common factors to link \( \frac{dB}{d\omega} \) with \( B \) and \( \omega\).

Isolate \(\frac{dB}{d\omega}\)

From the simplified equation, solve for \( \frac{dB}{d\omega} \):\[ \frac{dB}{d\omega} = \frac{m}{q} \cdot \frac{\omega R^{2}}{c^{2}} \left[1-(\omega R)^{2}/c^{2}\right]^{-3/2}.\]

Express in the required form

Use the simplified expression and the original equation to express: \[ \frac{dB}{d\omega} = \frac{B \cdot (\omega R)^{2}}{\omega}.\] Finally, relate \(dB\) and \(d\omega\) in the required forms.

Key concepts

relativistic mechanics

Relativistic mechanics is crucial for understanding how particles behave at speeds close to the speed of light. In these situations, the classical Newtonian mechanics no longer fully apply, and Einstein's theory of relativity becomes necessary.
Relativistic mechanics considers how time, length, and mass change for objects moving at high velocities. The famous equation \(E = mc^2\) shows that mass and energy are interchangeable.
The synchrotron equation incorporates relativistic effects through the term \([1-(\omega R)^{2} / c^{2}]^{-1 / 2}\). This factor adjusts for the fact that as a particle's speed approaches the speed of light, its effective mass appears to increase, affecting its motion in a magnetic field.
Understanding these concepts helps us grasp why particles in synchrotrons behave differently at high speeds compared to low speeds.

angular frequency

Angular frequency (\(\omega\)) describes how rapidly an object rotates or oscillates. It is related to the period of rotation or oscillation and is measured in radians per second.
In the context of the synchrotron equation, the angular frequency \(\omega\) represents the rate at which a charged particle travels around the circular path within the synchrotron.
The relation between \(\omega\) and other quantities in the equation shows how changing the particle's speed affects its trajectory and the magnetic field required to maintain that path.
Mathematically, angular frequency is related to the rotational frequency \(f\) by the formula \(\omega = 2\pi f\). This is essential for understanding how adjustments to \(\omega\) influence the behavior of the entire system.

magnetic field strength

Magnetic field strength \(B\) is a measure of the intensity of the magnetic field in a given region. It is crucial for controlling the path of charged particles in devices like synchrotrons.
In the synchrotron equation, \(B\) interacts with the charge \(q\) and the angular frequency \(\omega\) to dictate the particle's motion. A stronger magnetic field exerts a greater force on the particle, affecting its angular frequency and orbit radius.
The interplay between \(B\) and \(\omega\) is key to maintaining the particle's path. Adjusting the magnetic field can compensate for changes in the particle's speed, ensuring it remains in a controlled trajectory.
Understanding magnetic field strength helps in designing and operating synchrotrons effectively.

differentiation in calculus

Differentiation in calculus involves finding the rate at which a quantity changes. It is essential for solving problems related to changing physical quantities.
In the given problem, differentiation helps us understand how changes in the magnetic field \(dB\) relate to changes in the angular frequency \(d\omega\).
By differentiating both sides of the synchrotron equation, we apply the product rule and chain rule to find the relationship between these variables. This process involves expressing the rate of change of one variable in terms of others.
Differentiation is a powerful tool in physics and engineering, allowing us to model and predict complex systems' behavior accurately.