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Zorluk: OrtaConduction of Electricity Through Gases and Cathode Rays

In a cathode-ray experiment, a beam of electrons passes undeflected through mutually perpendicular electric and magnetic fields. If the electric field intensity between the deflection plates is 4.0×104 V m−14.0 \times 10^{4}\text{ V m}^{-1} and the magnetic flux density is 2.0×10−3 T2.0 \times 10^{-3}\text{ T}, what is the velocity of the electrons in the cathode-ray beam?

  1. 2.0×107 m s−12.0 \times 10^{7}\text{ m s}^{-1}Cevap
  2. B
    5.0×10−8 m s−15.0 \times 10^{-8}\text{ m s}^{-1}
  3. C
    8.0×101 m s−18.0 \times 10^{1}\text{ m s}^{-1}
  4. D
    6.0×107 m s−16.0 \times 10^{7}\text{ m s}^{-1}

Cevap

The velocity of the electrons in the cathode-ray beam is 2.0×107 m s−12.0 \times 10^{7}\text{ m s}^{-1}.
For cathode rays passing undeflected through perpendicular electric (EE) and magnetic (BB) fields, the electric force eEeE pulling electrons toward the positive plate is exactly balanced by the magnetic force evBevB. Equating eE=evBeE = evB leads directly to v=EBv = \frac{E}{B}. Substituting E=4.0×104 V m−1E = 4.0 \times 10^{4}\text{ V m}^{-1} and B=2.0×10−3 TB = 2.0 \times 10^{-3}\text{ T} gives v=4.0×1042.0×10−3=2.0×107 m s−1v = \frac{4.0 \times 10^{4}}{2.0 \times 10^{-3}} = 2.0 \times 10^{7}\text{ m s}^{-1}.

Adım Adım Çözüm

1
Identify the equilibrium condition for an undeflected electron beam in crossed fields.
The electric force Fe=eEF_e = eE balances the magnetic Lorentz force Fb=evBF_b = evB, so eE=evBeE = evB.
When cathode rays pass undeflected, the net transverse force acting on each electron is zero.
2
Rearrange the equilibrium equation to solve for velocity vv.
v=EBv = \frac{E}{B}
Canceling the elementary charge ee from both sides isolates electron speed as a function of field strengths.
3
Substitute the given values into the velocity equation.
v=4.0×104 V m−12.0×10−3 T=2.0×107 m s−1v = \frac{4.0 \times 10^{4}\text{ V m}^{-1}}{2.0 \times 10^{-3}\text{ T}} = 2.0 \times 10^{7}\text{ m s}^{-1}
Executing the division yields the velocity of cathode rays.

Anahtar Kavram

Velocity selector principle in crossed electric and magnetic fields (J.J. Thomson cathode ray velocity determination)
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Konular

Conduction of Electricity Through Gases and Cathode Rays

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