The electric current in an X-ray tube (from the target to the filament) operating at `40 kV `is `10 mA`. Assume that on an average, 1% of the total kinetic energy of the electrons hitting the target are converted into X-rays (a) what is the total power emitted as X-rays and (b) how much heat is produced in the target every second?

Explanation

To solve the problem step by step, we need to find the total power emitted as X-rays and the amount of heat produced in the target every second.

### Step 1: Convert Given Values
1. **Voltage (V)**: The operating voltage is given as \( 40 \, \text{kV} \).
\[ V = 40 \times 10^3 \, \text{V} = 40000 \, \text{V} \]

2. **Current (I)**: The current is given as \( 10 \, \text{mA} \).
\[ I = 10 \times 10^{-3} \, \text{A} = 0.01 \, \text{A} \]

### Step 2: Calculate the Total Charge (Q) per Second
The current (I) is defined as the charge (Q) per unit time (t). For \( t = 1 \, \text{s} \):
\[ Q = I \times t = 0.01 \, \text{A} \times 1 \, \text{s} = 0.01 \, \text{C} \]

### Step 3: Calculate the Number of Electrons (n)
The charge of a single electron (e) is approximately \( 1.6 \times 10^{-19} \, \text{C} \). The number of electrons hitting the target can be calculated using:
\[ n = \frac{Q}{e} = \frac{0.01 \, \text{C}}{1.6 \times 10^{-19} \, \text{C}} \approx 6.25 \times 10^{16} \, \text{electrons} \]

### Step 4: Calculate the Kinetic Energy (KE) of One Electron
The kinetic energy (KE) of an electron accelerated through a potential difference (V) is given by:
\[ \text{KE} = e \times V = (1.6 \times 10^{-19} \, \text{C}) \times (40000 \, \text{V}) = 6.4 \times 10^{-15} \, \text{J} \]

### Step 5: Calculate the Total Kinetic Energy (TKE)
The total kinetic energy of all the electrons hitting the target is:
\[ \text{TKE} = n \times \text{KE} = (6.25 \times 10^{16}) \times (6.4 \times 10^{-15} \, \text{J}) \approx 400 \, \text{J} \]

### Step 6: Calculate the Power Emitted as X-rays
Given that 1% of the total kinetic energy is converted into X-rays:
\[ \text{Power}_{\text{X-rays}} = 0.01 \times \text{TKE} = 0.01 \times 400 \, \text{J} = 4 \, \text{J} \]

### Step 7: Calculate the Heat Produced in the Target
The heat produced in the target is the remaining energy after accounting for the energy converted into X-rays:
\[ \text{Heat} = \text{TKE} - \text{Power}_{\text{X-rays}} = 400 \, \text{J} - 4 \, \text{J} = 396 \, \text{J} \]

### Final Answers
(a) The total power emitted as X-rays is \( 4 \, \text{J} \).
(b) The heat produced in the target every second is \( 396 \, \text{J} \).