Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

문제 36

37장, 문제 36

In one of Thomson’s experiments he placed a thin metal foil in the electron beam and measured its temperature rise. Consider a cathode-ray tube in which electrons are accelerated through a 2000 V potential difference, then strike a 10 mg copper foil. What is the electron-beam current if the foil temperature rises 6.0°C in 10 s? Assume no loss of energy by radiation or other means. The specific heat of copper is 385 J/kg K .

검증된 단계별 안내

Step 1: Calculate the energy gained by each electron as it is accelerated through the potential difference. The energy is given by the equation:

E = q V

, where

q

is the charge of an electron (1.6 × 10⁻¹⁹ C) and

V

is the potential difference (2000 V).

Step 2: Determine the total heat energy absorbed by the copper foil using the formula:

Q = m c ΔT

, where

m

is the mass of the foil (10 mg = 10 × 10⁻³ g = 10⁻⁵ kg),

c

is the specific heat of copper (385 J/kg·K), and

ΔT

is the temperature rise (6.0°C).

Step 3: Calculate the total energy delivered by the electron beam in 10 seconds. Since all the energy from the electrons is transferred to the foil, the total energy delivered is equal to the heat energy

Q

calculated in Step 2.

Step 4: Determine the number of electrons required to deliver this energy. Use the relationship:

n = \frac{Q}{E}

, where

E

is the energy per electron (calculated in Step 1).

Step 5: Calculate the electron-beam current using the formula:

I = \frac{n}{t} q

, where

n

is the number of electrons (from Step 4),

t

is the time (10 s), and

q

is the charge of an electron.

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주요 개념

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Electron Beam Current

Electron beam current refers to the flow of electrons through a conductor, measured in amperes (A). In this context, it can be calculated using the relationship between the energy transferred to the copper foil and the charge of the electrons. The current can be determined by dividing the total charge that strikes the foil by the time interval during which the charge flows.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). For copper, this value is 385 J/kg K, meaning that it takes 385 joules of energy to increase the temperature of one kilogram of copper by one degree. This concept is crucial for calculating the energy absorbed by the foil and subsequently determining the current.

Energy Transfer in Electrical Systems

Energy transfer in electrical systems involves the conversion of electrical energy into other forms, such as thermal energy. In this scenario, the kinetic energy of the accelerated electrons is converted into heat when they collide with the copper foil, causing its temperature to rise. The total energy transferred can be calculated using the specific heat formula, which relates the mass, specific heat capacity, and temperature change.