Question

An electron moves in a circular trajectory with radius \(r_{\mathrm{i}}\) in a constant magnetic field. What is the final radius of the trajectory when the magnetic field is doubled? a) \(\frac{r_{i}}{4}\) b) \(\frac{r_{i}}{2}\) c) \(r_{i}\) d) \(2 r_{i}\) e) \(4 r_{\mathrm{i}}\)

Short answer

Question: When the magnetic field in which an electron is moving in a circular trajectory is doubled, the final radius of the trajectory is: a) Twice the initial radius b) Half the initial radius c) The same as the initial radius d) Four times the initial radius Answer: b) Half the initial radius

Step-by-step solution

Setting the magnetic force and centripetal force equal to each other

We have two equations: \(F = qvB\) (magnetic force) and \(F = \frac{mv^2}{r}\) (centripetal force). Since the magnetic force is providing the centripetal force for the electron, we can set these two equations equal to each other: $$ \Rightarrow qvB = \frac{mv^2}{r} $$

Solving for the radius in terms of the magnetic field

Now, we can solve for the radius \(r\) in terms of the magnetic field \(B\). Rearrange the equation as follows: $$ \Rightarrow r = \frac{mv}{qB} $$

Doubling the magnetic field

We are given that the magnetic field is doubled. Let \(r_i\) be the initial radius and \(r_f\) be the final radius. We can write two equations: $$ \begin{cases}r_i = \frac{mv}{qB_i}\\ r_f = \frac{mv}{q(2B_i)} \end{cases} $$

Finding the relationship between initial and final radii

Now, to find the relationship between \(r_i\) and \(r_f\), we can divide the second equation by the first equation: $$ \frac{r_f}{r_i} = \frac{\frac{mv}{q(2B_i)}}{\frac{mv}{qB_i}} = \frac{B_i}{2B_i} = \frac{1}{2} $$

Solving for the final radius

We have found the relationship between the initial and final radii, so we can now solve for the final radius \(r_f\) in terms of the initial radius \(r_i\): $$ \Rightarrow r_f = \frac{1}{2} r_i $$ The final radius of the trajectory when the magnetic field is doubled is \(\frac{r_i}{2}\). Therefore, the correct answer is option (b).

Key concepts

Centripetal Force

Centripetal force is a crucial concept when dealing with circular motion. It is the force that keeps an object moving along a curved path, directed towards the center of curvature. When an electron moves in a circular path within a magnetic field, the centripetal force is provided by the magnetic force. This is why their equations can be set equal to one another: \( F = qvB = \frac{mv^2}{r} \).

• \(F\) is the force acting towards the center.
• \(m\) is the mass of the electron.
• \(v\) is the speed of the electron.
• \(r\) is the radius of the trajectory.
• \(q\) is the charge of the electron.
• \(B\) is the magnetic field strength.

By understanding centripetal force, you can see why increasing the magnetic field affects the radius. Doubling the magnetic field increases the force acting on the electron, which pulls it more strongly toward the center, reducing the radius of the path.

Electron Trajectory

An electron trajectory refers to the path that an electron takes through space. When an electron is in a magnetic field, its trajectory is typically a circular path. The direction and radius of this path depend on several factors, including the velocity of the electron and the strength of the magnetic field.If the magnetic field increases, as in our initial exercise, electrons will tend to have a smaller radius of curvature as they are pulled more tightly into their circular paths. The key equation here is: \[ r = \frac{mv}{qB} \]This showcases that the trajectory's radius \( r \) decreases when the magnetic field \( B \) increases, holding everything else constant. When the magnetic field is doubled, the radius becomes half, demonstrating the inverse relationship between them. Thus, understanding this principle is paramount when predicting how an electron will move in varying magnetic fields.

Magnetic Force

Magnetic force affects charged particles such as electrons moving through magnetic fields. It acts perpendicular to both the direction of the magnetic field and the velocity of the particle. This is central to the creation of a circular path for the electron.The formula for magnetic force is: \[ F = qvB \] - \(F\) represents the magnetic force.- \(q\) is the charge of the particle.- \(v\) is the velocity of the particle.- \(B\) is the strength of the magnetic field.The magnetic force is key to maintaining the electron in motion along its curved path by providing the necessary centripetal force. When the magnetic field is strengthened, the force increases, causing the trajectory radius to shrink as seen in the exercise. Understanding the effects of magnetic force enhances comprehension of electron behavior in different field strengths.