The ratio of specific charge of a proton and an `alpha`-particle is

Options

A. `2:1`
B. `1:2`
C. `1:4`
D. `1:1`

Answer

To find the ratio of the specific charge of a proton to that of an alpha particle, we will follow these steps:

### Step 1: Understand Specific Charge
The specific charge of a particle is defined as the ratio of its charge (Q) to its mass (m):
\[ \text{Specific Charge} = \frac{Q}{m} \]

### Step 2: Calculate the Specific Charge of a Proton
The charge of a proton (Q_p) is:
\[ Q_p = +1.6 \times 10^{-19} \text{ C} \]
The mass of a proton (m_p) is:
\[ m_p = 1.66 \times 10^{-27} \text{ kg} \]
Now, we can calculate the specific charge of the proton:
\[ \text{Specific Charge of Proton} = \frac{Q_p}{m_p} = \frac{1.6 \times 10^{-19}}{1.66 \times 10^{-27}} \]

### Step 3: Calculate the Specific Charge of an Alpha Particle
An alpha particle consists of 2 protons and 2 neutrons, and it has a charge of:
\[ Q_{\alpha} = +2 \times 1.6 \times 10^{-19} \text{ C} = 3.2 \times 10^{-19} \text{ C} \]
The mass of an alpha particle (m_α) is approximately:
\[ m_{\alpha} = 4 \text{ AMU} = 4 \times 1.66 \times 10^{-27} \text{ kg} = 6.64 \times 10^{-27} \text{ kg} \]
Now, we can calculate the specific charge of the alpha particle:
\[ \text{Specific Charge of Alpha Particle} = \frac{Q_{\alpha}}{m_{\alpha}} = \frac{3.2 \times 10^{-19}}{6.64 \times 10^{-27}} \]

### Step 4: Calculate the Ratio of Specific Charges
Now, we need to find the ratio of the specific charge of a proton to that of an alpha particle:
\[ \text{Ratio} = \frac{\text{Specific Charge of Proton}}{\text{Specific Charge of Alpha Particle}} = \frac{\frac{1.6 \times 10^{-19}}{1.66 \times 10^{-27}}}{\frac{3.2 \times 10^{-19}}{6.64 \times 10^{-27}}} \]

### Step 5: Simplify the Ratio
This can be simplified as follows:
\[ \text{Ratio} = \frac{1.6 \times 10^{-19} \times 6.64 \times 10^{-27}}{3.2 \times 10^{-19} \times 1.66 \times 10^{-27}} \]
Notice that \(1.6 \times 10^{-19}\) cancels with \(3.2 \times 10^{-19}\) and \(1.66 \times 10^{-27}\) cancels with \(6.64 \times 10^{-27}\):
\[ \text{Ratio} = \frac{1}{2} \]

### Final Result
Thus, the ratio of the specific charge of a proton to that of an alpha particle is:
\[ \text{Ratio} = 2:1 \]