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

Options

A. (a) `2:1`
B. (b) `1:2`
C. (c) `1:4`
D. (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: Define Specific Charge
Specific charge is defined as the charge per unit mass, given by the formula:
\[ \text{Specific Charge} = \frac{Q}{m} \]
where \( Q \) is the charge and \( m \) is the mass.

### Step 2: Determine the Charge and Mass of a Proton
- The charge of a proton (\( Q_p \)) is \( +1 \) elementary charge, which can be expressed as \( +e \).
- The mass of a proton (\( m_p \)) is denoted as \( m_p \).

Thus, the specific charge of a proton (\( SC_p \)) can be calculated as:
\[ SC_p = \frac{Q_p}{m_p} = \frac{+e}{m_p} \]

### Step 3: Determine the Charge and Mass of an Alpha Particle
An alpha particle (\( \alpha \)) is essentially a helium nucleus, which consists of:
- 2 protons
- 2 neutrons

- The charge of an alpha particle (\( Q_{\alpha} \)) is \( +2e \) (since it has 2 protons).
- The mass of an alpha particle (\( m_{\alpha} \)) is approximately the sum of the masses of its constituents:
\[ m_{\alpha} = 2m_p + 2m_n \approx 4m_p \]
(assuming the mass of a neutron \( m_n \) is approximately equal to the mass of a proton).

### Step 4: Calculate the Specific Charge of the Alpha Particle
The specific charge of an alpha particle (\( SC_{\alpha} \)) can be calculated as:
\[ SC_{\alpha} = \frac{Q_{\alpha}}{m_{\alpha}} = \frac{+2e}{4m_p} = \frac{+e}{2m_p} \]

### Step 5: Find the Ratio of Specific Charges
Now, we can find the ratio of the specific charge of a proton to that of an alpha particle:
\[ \text{Ratio} = \frac{SC_p}{SC_{\alpha}} = \frac{\frac{+e}{m_p}}{\frac{+e}{2m_p}} = \frac{+e \cdot 2m_p}{+e \cdot m_p} = 2 \]

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

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