Crossed electric and magnetic fields have magnitudes of 4.65 V/m and 74.5 mT. Find the speed at which electrons moving perpendicular to both fields can travel through them without being deflected?
Crossed electric and magnetic fields have magnitudes of 4.65 V/m and 74.5 mT. Find the speed at which electrons moving perpendicular to both fields can travel through them without being deflected?
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![**Problem: Motion of Electrons in Crossed Fields**
**Description:**
In a region of space, crossed electric and magnetic fields are present. The magnitudes of these fields are given as:
- Electric field (E): 4.65 V/m
- Magnetic field (B): 74.5 mT (millitesla)
**Question:**
At what speed can electrons move perpendicular to both fields without being deflected?
**Solution Approach:**
To find the speed at which electrons can travel through the crossed electric and magnetic fields without deflection, use the condition for zero net force: the electric force must equal the magnetic force.
The relevant equation is:
\[ v = \frac{E}{B} \]
Where:
- \( v \) is the speed of the electrons
- \( E \) is the magnitude of the electric field (4.65 V/m)
- \( B \) is the magnitude of the magnetic field (74.5 mT converted to T for calculation)
Convert B to Tesla:
\[ 1 \text{ mT} = 10^{-3} \text{ T} \]
\[ B = 74.5 \text{ mT} = 74.5 \times 10^{-3} \text{ T} \]
Substitute known values into the formula to find \( v \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71da004f-23dc-47ac-9c54-84cf84b3724a%2Fe5d8fdc7-1a9e-4c2f-b309-e2a4d1c78b0a%2Fru8js5g_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Motion of Electrons in Crossed Fields**
**Description:**
In a region of space, crossed electric and magnetic fields are present. The magnitudes of these fields are given as:
- Electric field (E): 4.65 V/m
- Magnetic field (B): 74.5 mT (millitesla)
**Question:**
At what speed can electrons move perpendicular to both fields without being deflected?
**Solution Approach:**
To find the speed at which electrons can travel through the crossed electric and magnetic fields without deflection, use the condition for zero net force: the electric force must equal the magnetic force.
The relevant equation is:
\[ v = \frac{E}{B} \]
Where:
- \( v \) is the speed of the electrons
- \( E \) is the magnitude of the electric field (4.65 V/m)
- \( B \) is the magnitude of the magnetic field (74.5 mT converted to T for calculation)
Convert B to Tesla:
\[ 1 \text{ mT} = 10^{-3} \text{ T} \]
\[ B = 74.5 \text{ mT} = 74.5 \times 10^{-3} \text{ T} \]
Substitute known values into the formula to find \( v \).
Expert Solution
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Step 1: Formula
The electron will move undeflected when the force on the electron due to the electric field will be equal to the Lorrentz force. Hence,
Hence,
is the velocity of the electron such that it moves undeflected.
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