(iii) A disk of radius 2.5 cm has a surface charge density of 5.3mu C/m^2 on its upper face. What is the magnitude of the electric field produced by the disk at a point on its central axis at distance z=12cm from the disk? (4 mks)

To find the magnitude of the electric field produced by the disk at a point on its central axis at a distance \( z = 12 \) cm from the disk, we can use the formula for the electric field due to a uniformly charged disk:<br /><br />\[ E = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{z}{\sqrt{z^2 + R^2}}\right) \]<br /><br />where:<br />- \( E \) is the electric field,<br />- \( \sigma \) is the surface charge density,<br />- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \)),<br />- \( z \) is the distance from the disk along the central axis,<br />- \( R \) is the radius of the disk.<br /><br />Given:<br />- \( \sigma = 5.3 \, \mu\text{C/m}^2 = 5.3 \times 10^{-6} \, \text{C/m}^2 \),<br />- \( R = 2.5 \, \text{cm} = 0.025 \, \text{m} \),<br />- \( z = 12 \, \text{cm} = 0.12 \, \text{m} \).<br /><br />Substitute these values into the formula:<br /><br />\[ E = \frac{5.3 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} \left(1 - \frac{0.12}{\sqrt{0.12^2 + 0.025^2}}\right) \]<br /><br />Calculate the denominator:<br /><br />\[ 2 \times 8.85 \times 10^{-12} = 1.77 \times 10^{-11} \]<br /><br />Calculate the term inside the parentheses:<br /><br />\[ \sqrt{0.12^2 + 0.025^2} = \sqrt{0.0144 + 0.000625} = \sqrt{0.014925} \approx 0.1216 \]<br /><br />\[ \frac{0.12}{0.1216} \approx 0.987 \]<br /><br />So,<br /><br />\[ 1 - 0.987 = 0.013 \]<br /><br />Now, substitute back into the equation:<br /><br />\[ E = \frac{5.3 \times 10^{-6}}{1.77 \times 10^{-11}} \times 0.013 \]<br /><br />\[ E \approx 2.94 \times 10^{4} \, \text{N/C} \]<br /><br />Therefore, the magnitude of the electric field produced by the disk at a point on its central axis at a distance \( z = 12 \) cm from the disk is approximately \( 2.94 \times 10^4 \, \text{N/C} \).