In a television picture tube, electrons strike the screen after being accelerated from rest through a potential difference of 25000V. The speeds of the electrons are quite large, and for accurate calculations of the speeds, the effects of special relativity must be taken into account. Ignoring such effects, find the electron speed just before the electron strikes the screen.
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Accelerating Potential * e= 1/2 m v^2
In a television picture tube, electrons strike the screen after being accelerated from rest through a potential difference of 33000 V. The speeds of the electrons are quite large, and for accurate calculations of the speeds, the effects of special relativity must be taken into account. Ignoring such effects, find the electron speed just before the electron strikes the screen.
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To determine the speed of the electrons just before they strike the screen, you need to apply the principles of classical mechanics. In this scenario, the effects of special relativity are not considered. Here's how you can approach the problem:
1. Identify the given information:
- Potential difference (V) = 25000V
- Rest mass of an electron (m) = 9.1 x 10^-31 kg
- Charge of an electron (e) = -1.6 x 10^-19 C
2. Determine the work done on the electrons:
The electron gains energy as it is accelerated through the potential difference. The work done (W) on the electron is given by the formula:
W = qV
where q represents the charge of the electron (e) and V is the potential difference.
Substituting the values:
W = (-1.6 x 10^-19 C) x (25000V) = -4.0 x 10^-15 J (negative sign due to the electron's charge)
3. Use the work-energy theorem:
The change in kinetic energy (∆KE) is equal to the work done (W). The change in kinetic energy can be expressed as:
∆KE = KEfinal - KEinitial
Since the electron starts from rest, the initial kinetic energy (KEinitial) is zero. Therefore:
∆KE = KEfinal - 0
Simplifying this equation gives:
KEfinal = ∆KE = W
4. Apply the kinetic energy formula:
The kinetic energy (KE) of an object can be calculated using the formula:
KE = (1/2)mv^2
where m is the mass of the electron and v is its velocity.
Set up the equation using the values:
-4.0 x 10^-15 J = (1/2)(9.1 x 10^-31 kg)(v^2)
5. Solve for the velocity (v):
Rearranging the equation gives:
8.0 x 10^-15 J = 9.1 x 10^-31 kg * v^2
Divide both sides by (9.1 x 10^-31 kg) to isolate v^2:
v^2 = (8.0 x 10^-15 J) / (9.1 x 10^-31 kg)
Take the square root of both sides to find v:
v ≈ √[(8.0 x 10^-15 J) / (9.1 x 10^-31 kg)]
6. Calculate the result:
Plug the values into a calculator to find the approximate velocity of the electrons just before they strike the screen.
It's important to note that this calculation assumes classical mechanics and neglects the effects of special relativity. For more accurate calculations involving high speeds, special relativity must be considered.