In an old-fashioned television tube, an electron ) starting from rest experiences a force of 4.0 x 10^-15 N over a distance of 50 cm. Ignoring the relativistic effects, the final speed of the electron is:

Question

Quizplus · Accepted Answer

We can start by using the work-energy principle, which tells us that the work done on an object equals its change in kinetic energy:

$$W=\Delta K$$

In this case, the electron starts from rest, so its initial kinetic energy is zero. The work done on the electron is given by:

$$W=F\cdot d=(4.0	imes 10^{-15}	ext{ N})(0.5	ext{ m})=2.0	imes 10^{-15}	ext{ J}$$

So the final kinetic energy of the electron is:

$$\Delta K=2.0	imes 10^{-15}	ext{ J}$$

Since the electron is moving at relativistic speeds, we need to use the relativistic kinetic energy equation:

$$K=(\gamma-1)mc^2$$

where $\gamma$ is the Lorentz factor, $m$ is the mass of the electron, and $c$ is the speed of light. The Lorentz factor is given by:

$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

where $v$ is the speed of the electron. We can rearrange this equation to solve for $v$:

$$v=c\sqrt{1-\frac{1}{\gamma^2}}$$

Since we know the change in kinetic energy and the mass of the electron, we can set up an equation relating the final kinetic energy to the final speed:

$$(\gamma-1)mc^2=\Delta K=\frac{1}{2}mv^2$$

Simplifying this equation and solving for $v$, we get:

$$v=c\sqrt{1-\frac{2}{\gamma}}$$

Now we just need to plug in the values given in the problem. The mass of the electron is approximately $9.1	imes 10^{-31}$ kg. The speed of light is approximately $3.0	imes 10^8$ m/s. We already found that $\Delta K=2.0	imes 10^{-15}$ J. So we just need to solve for $\gamma$:

$$(\gamma-1)(9.1	imes 10^{-31}	ext{ kg})(3.0	imes 10^8	ext{ m/s})^2=2.0	imes 10^{-15}	ext{ J}$$

Solving for $\gamma$, we get:

$$\gamma=1.0010986$$

Finally, we can plug this value into the equation for $v$:

$$v=(3.0	imes 10^8	ext{ m/s})\sqrt{1-\frac{2}{1.0010986}}\approx 6.6	imes 10^6	ext{ m/s}$$

So the answer is (B).

In an Old-Fashioned Television Tube, an Electron ) Starting from Rest