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You and your team are tasked with designing an "electron gun" that operates in a vacuum chamber, the purpose of which is to direct a beam of electrons toward a tiny metallic plate in order to heat it. The electrons in the beam have a speed of \(3.50 \times 10^{7} \mathrm{m} / \mathrm{s}\) and travel in the positive \(x\) direction through the center of a set of deflecting plates (a parallel-plate capacitor) that sets up a uniform electric field in the region between the plates. The target is located \(22.0 \mathrm{cm}\) along the \(x\) -axis from the trailing edge of plates (i.e., the edge closest to the target), and \(11.5 \mathrm{cm}\) above the horizontal (i.e., in the \(+y\) direction). The length of the plates (in the \(x\) direction) is \(2.50 \mathrm{cm}\) (a) In which direction should the electric field between the plates point in order to deflect the electrons towards the target? (b) To what magnitude should you set the electric field so that the electron beam hits the target? (c) After successfully striking the target using your results from (a) and (b), you realized that the target is not heating up to the required temperature. Since the degree of heating depends on the speed of the electrons, you increase the electron speed to \(5.20 \times 10^{7} \mathrm{m} / \mathrm{s} .\) With the electric field setting from (b), will the electrons still be on target? If not, to what value should you set the electric field?

Step by step solution

01

Understanding Electron Deflection

Electrons are negatively charged, meaning they are deflected in the direction opposite to that of the electric field. To deflect them upwards towards the target, the electric field must point downwards.

02

Determine Electric Field Direction

Since the electrons need to hit a target 11.5 cm above their initial path, the electric field should point in the negative y-direction to push the electrons upward.

03

Calculate Time in Electric Field Region

The time, \(t_{plates}\), electrons take to travel the distance of the plates (2.50 cm) can be calculated using the speed along the \(x\)-axis. \[ t_{plates} = \frac{0.025 \, ext{m}}{3.50 \times 10^{7} \, ext{m/s}} = 7.14 \times 10^{-10} \, ext{s} \]

04

Compute Necessary Vertical Deflection

Electrons must be deflected by 11.5 cm. Using the equation for motion under constant acceleration, we find the required field strength. First, use the full distance electrons need to travel: \(d_y = 0.115 \, ext{m}\). The travel time for the whole gate leading to target is: \[ t_{total} = \frac{0.22 \, ext{m}}{3.50 \times 10^{7} \, ext{m/s}} = 6.29 \times 10^{-9} \, ext{s} \]

05

Find Acceleration from Electric Field

The acceleration required to move vertically: \[ a_y = \frac{2d_y}{t_{total}^2} \] Given acceleration uses electrons: \[ qE = ma_y \rightarrow E = \frac{m \cdot a_y}{-e} \] where \(e\) is the charge of an electron \(1.60 \times 10^{-19} \, ext{C}\) and mass \(9.11 \times 10^{-31} \, ext{kg}\). Use known values to compute magnitude of \(E\).

06

Recalculate for Increased Speed

Changes in projectile speed require recalculation. New speed \(v=5.20 \times 10^{7} \, ext{m/s}\), \(t_{total}' = \frac{0.22 \, ext{m}}{5.20 \times 10^{7} \, ext{m/s}} \). Find new acceleration and again solve for \(E\).

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Key Concepts

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

Deflection of Electrons

When electrons pass through an electric field, they are deflected due to their negative charge. The direction of this deflection is opposite to the direction of the electric field. Imagine pushing a ball uphill; it goes against your push. That's how electrons behave, moving in the opposite path to where the field points.
To hit a target located above their path, the electric field needs to point downward. This downward field will push the electrons up towards the target. Think of it like a trampoline pushing you in the opposite direction.

Electric Field

An electric field exerts force on any charged particle in its vicinity. The field is uniform between parallel plates, meaning the force is consistent. Imagine standing in a gentle wind that pushes you evenly in one direction.
The direction of the field is crucial in guiding the electrons. In our case, the field needs to point downwards to deflect negatively charged electrons upward. The strength of the field, or its magnitude, determines how much force is applied. This force influences how much the electron's path bends, so getting the magnitude just right ensures precision in hitting a target.

Parallel-Plate Capacitor

A parallel-plate capacitor is made of two conductive plates with space in between. When connected to a voltage, this setup produces a uniform electric field between the plates.
For our electron gun, placing these plates along the electron's path can steer where they go. The uniform field ensures a predictable, steady force is applied, making it easier to calculate the deflection needed.

• Plates create a field that is consistent, meaning calculations for deflection are simplified.
• The spacing and length of the plates can influence the field's strength and, thus, the extent of deflection.

Electron Speed and Kinetics

The speed of an electron affects how long it remains in the electric field and how much it is deflected. Faster electrons spend less time between the plates, meaning they are influenced less by the electric field.
When speed increases, as with our scenario from 3.50 to 5.20 times 10鈦� m/s, the time to cross the plates decreases. This requires recalculating the electric field's strength for the desired deflection.

• Higher speeds necessitate adjustments in electric field strength to maintain trajectory.
• Kinetic energy of electrons, determined by speed, impacts how effectively they can heat a target upon impact.

Understanding these concepts ensures precise tuning of both speed and field, enabling accurate targeting.