Use the work-energy theorem to find the force required to accelerate an electron `( m = 9.11 xx 10^(-31) kg)` from rest to a speed of `1.50xx10^(7)`m/s in a distance of 0.0125 m.

Options

A. `8.20xx10^(-15)N`
B. `8.20xx10^(-17)N`
C. `5.47xx10^(-22)N`
D. `1.64xx10^(-14)N`

Answer

To find the force required to accelerate an electron from rest to a speed of \(1.50 \times 10^7 \, \text{m/s}\) over a distance of \(0.0125 \, \text{m}\) using the work-energy theorem, we can follow these steps:

### Step 1: Understand the Work-Energy Theorem
The work-energy theorem states that the work done by the net force acting on an object is equal to the change in its kinetic energy. Mathematically, it can be expressed as:
\[ W = \Delta KE \]
where \(W\) is the work done and \(\Delta KE\) is the change in kinetic energy.

### Step 2: Calculate the Change in Kinetic Energy
The change in kinetic energy can be calculated using the formula:
\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]
Since the electron starts from rest, its initial kinetic energy \(KE_{\text{initial}} = 0\). The final kinetic energy is given by:
\[ KE_{\text{final}} = \frac{1}{2} m v^2 \]
where \(m\) is the mass of the electron and \(v\) is its final speed.

### Step 3: Substitute the Values
Given:
- Mass of the electron, \(m = 9.11 \times 10^{-31} \, \text{kg}\)
- Final speed, \(v = 1.50 \times 10^7 \, \text{m/s}\)

Now, substituting these values into the kinetic energy formula:
\[ KE_{\text{final}} = \frac{1}{2} \times (9.11 \times 10^{-31}) \times (1.50 \times 10^7)^2 \]

### Step 4: Calculate the Kinetic Energy
Calculating \(v^2\):
\[ (1.50 \times 10^7)^2 = 2.25 \times 10^{14} \]
Now substituting back:
\[ KE_{\text{final}} = \frac{1}{2} \times (9.11 \times 10^{-31}) \times (2.25 \times 10^{14}) = 1.024875 \times 10^{-16} \, \text{J} \]

### Step 5: Calculate the Work Done
Since the initial kinetic energy is zero, the work done \(W\) is equal to the final kinetic energy:
\[ W = KE_{\text{final}} = 1.024875 \times 10^{-16} \, \text{J} \]

### Step 6: Relate Work Done to Force
The work done can also be expressed in terms of force and distance:
\[ W = F \cdot d \]
where \(F\) is the force and \(d\) is the distance over which the force is applied. Rearranging gives:
\[ F = \frac{W}{d} \]

### Step 7: Substitute the Values to Find Force
Given the distance \(d = 0.0125 \, \text{m}\):
\[ F = \frac{1.024875 \times 10^{-16}}{0.0125} \]
Calculating this gives:
\[ F = 8.199 \times 10^{-15} \, \text{N} \]

### Final Answer
The force required to accelerate the electron is approximately:
\[ F \approx 8.20 \times 10^{-15} \, \text{N} \]

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