In a cyclotron, if a deuteron can gain an energy of 40 MeV, then a proton can gain an energy of

Options

A. 40 MeV
B. 60 MeV
C. 20 MeV
D. 80 MeV

Answer

To solve the problem, we need to determine how much energy a proton can gain in a cyclotron, given that a deuteron can gain an energy of 40 MeV.

### Step-by-Step Solution:

1. **Understanding the Cyclotron**:
- In a cyclotron, charged particles (like protons and deuterons) gain energy due to the electric field and are kept in circular motion by a magnetic field.

2. **Kinetic Energy Formula**:
- The kinetic energy (KE) of a charged particle in a cyclotron can be expressed as:
\[ KE = \frac{q^2 B^2 r^2}{2m} \]
- Where:
- \( q \) = charge of the particle
- \( B \) = magnetic field strength
- \( r \) = radius of the circular path
- \( m \) = mass of the particle

3. **Kinetic Energy for Deuteron**:
- For a deuteron (which has a charge \( q_D \) and mass \( m_D \)):
\[ KE_D = \frac{q_D^2 B^2 r^2}{2m_D} \]
- Given that \( KE_D = 40 \, \text{MeV} \).

4. **Kinetic Energy for Proton**:
- For a proton (which has a charge \( q_P \) and mass \( m_P \)):
\[ KE_P = \frac{q_P^2 B^2 r^2}{2m_P} \]

5. **Comparing Energies**:
- We can compare the kinetic energies of the proton and deuteron:
\[ \frac{KE_P}{KE_D} = \frac{q_P^2 B^2 r^2 / (2m_P)}{q_D^2 B^2 r^2 / (2m_D)} \]
- Simplifying this gives:
\[ \frac{KE_P}{KE_D} = \frac{q_P^2 m_D}{q_D^2 m_P} \]

6. **Charge and Mass Values**:
- The charge of both the proton and deuteron is the same (both have charge +1 in elementary charge units).
- The mass of the deuteron \( m_D \) is approximately twice the mass of the proton \( m_P \):
\[ m_D = 2m_P \]

7. **Substituting Values**:
- Since \( q_P = q_D \), we can cancel the charges:
\[ \frac{KE_P}{KE_D} = \frac{m_D}{m_P} = \frac{2m_P}{m_P} = 2 \]
- Thus, we find:
\[ KE_P = 2 \times KE_D \]

8. **Calculating Proton Energy**:
- Now substituting the known value of \( KE_D \):
\[ KE_P = 2 \times 40 \, \text{MeV} = 80 \, \text{MeV} \]

### Final Answer:
The energy that a proton can gain in the cyclotron is **80 MeV**.

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