Electron in Aligned Electric and Magnetic Fields
Question: A uniform electric field and a uniform magnetic field act along the same direction in a certain region. An electron is projected in this region such that its velocity is pointed along the direction of the fields. Then the electron (2011):
Detailed Step-by-Step Explanation
Step 1: Identify the Fields and the Electron's Motion
In the given region, both the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are aligned in the same direction.
The electron is projected with a velocity \( \vec{v} \) that is parallel to these fields.
Step 2: Forces Acting on the Electron
The force on a charged particle moving in electric and magnetic fields is given by the Lorentz force:
\[
\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})
\]
where \( q \) is the charge of the particle.
For an electron, \( q = -e \) (with \( e \approx 1.6 \times 10^{-19}\,\text{C} \)).
Step 3: Analyze the Magnetic Force
The magnetic part of the force is:
\[
\vec{F}_B = q (\vec{v} \times \vec{B})
\]
Since the electron's velocity \( \vec{v} \) is parallel to the magnetic field \( \vec{B} \), the cross product \( \vec{v} \times \vec{B} \) is zero. Thus:
\[
\vec{F}_B = 0
\]
Step 4: Analyze the Electric Force
The electric part of the force is:
\[
\vec{F}_E = q \vec{E}
\]
Since the electric field and the electron's velocity are in the same direction, the force \( \vec{F}_E \) will act along the direction of \( \vec{E} \).
However, note that the electron carries a negative charge. Therefore, the electric force will be in the direction opposite to the electric field.
Step 5: Effect on the Electron's Motion
The net force on the electron is solely due to the electric field:
\[
\vec{F} = q \vec{E}
\]
Since this force is opposite to the electron's velocity (which is in the same direction as \( \vec{E} \)), the electron will experience a deceleration. In other words, its speed will decrease.
Final Answer: The electron's speed will decrease, which corresponds to option (b).
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