To determine the electric field created by a proton at a distance of 0.1 mm away from it, we can use Coulomb's law.
### Step 1: Identify the Known Values
1. Charge of a proton ([tex]\( q \)[/tex]) = [tex]\( 1.6 \times 10^{-19} \)[/tex] Coulombs
2. Distance from the proton ([tex]\( r \)[/tex]) = 0.1 mm = 0.1 \times 10^{-3} meters = [tex]\( 1 \times 10^{-4} \)[/tex] meters
3. Coulomb's constant ([tex]\( k \)[/tex]) = [tex]\( 8.99 \times 10^9 \)[/tex] N m²/C²
### Step 2: Write the Formula for Electric Field
The electric field [tex]\( E \)[/tex] created by a point charge is given by:
[tex]\[ E = \frac{k \cdot |q|}{r^2} \][/tex]
### Step 3: Substitute the Values
Substitute the known values into the formula:
[tex]\[ E = \frac{8.99 \times 10^9 \cdot 1.6 \times 10^{-19}}{(1 \times 10^{-4})^2} \][/tex]
### Step 4: Simplify the Denominator
First, simplify the denominator:
[tex]\[ (1 \times 10^{-4})^2 = 1 \times 10^{-8} \text{ meters}^2 \][/tex]
### Step 5: Perform the Division
Next, substitute this into the formula:
[tex]\[ E = \frac{8.99 \times 10^9 \cdot 1.6 \times 10^{-19}}{1 \times 10^{-8}} \][/tex]
### Step 6: Combine and Simplify the Numerical Terms
[tex]\[ E = \frac{8.99 \times 10^9 \cdot 1.6 \times 10^{-19}}{10^{-8}} \][/tex]
[tex]\[ E = 8.99 \times 1.6 \times 10^{9 + (-19) + 8} \][/tex]
[tex]\[ E = 8.99 \times 1.6 \times 10^{-2} \][/tex]
Perform the multiplication of the constants:
[tex]\[ E = 14.384 \times 10^{-2} \][/tex]
[tex]\[ E = 0.14384 \][/tex]
Therefore, the electric field created by a proton at a distance of 0.1 mm away from it is [tex]\( 0.14384 \)[/tex] N/C.
