Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Certainly! Let's walk through the process of calculating the electric field intensity at a point due to a given point charge step-by-step.
### Given Data
1. Charge (q): The charge is [tex]\(0.009 \mu C\)[/tex].
- Convert microcoulombs to coulombs: [tex]\(0.009 \mu C = 0.009 \times 10^{-6} C = 0.009 \times 10^{-6} C\)[/tex].
2. Permittivity of Free Space (ε₀): The permittivity of free space is [tex]\(8.854 \times 10^{-12} \, F/m\)[/tex].
3. Coordinates (x, y, z): The point at which we need to find the electric field intensity is [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex].
### Step-by-Step Solution
#### Step 1: Calculate the Distance from the Origin
- The charge is placed at the origin, [tex]\((0, 0, 0) \, m\)[/tex].
- The point at which we need to calculate the electric field is [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex].
- The distance [tex]\(r\)[/tex] from the origin to the point is calculated using the distance formula:
[tex]\[ r = \sqrt{(\sqrt{2} - 0)^2 + (\sqrt{7} - 0)^2 + (0 - 0)^2} \][/tex]
Simplifying inside the square root:
[tex]\[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{7})^2} \][/tex]
[tex]\[ r = \sqrt{2 + 7} \][/tex]
[tex]\[ r = \sqrt{9} \][/tex]
[tex]\[ r = 3 \, m \][/tex]
#### Step 2: Electric Field Intensity Calculation
- The electric field intensity [tex]\(E\)[/tex] due to a point charge is given by Coulomb's Law:
[tex]\[ E = \frac{k |q|}{r^2} \][/tex]
where [tex]\( k \)[/tex] (constant) can be expressed as [tex]\( \frac{1}{4 \pi \epsilon_0} \)[/tex], and [tex]\(\epsilon_0\)[/tex] is the permittivity of free space.
Substituting the known values:
[tex]\[ k = \frac{1}{4 \pi \epsilon_0} = \frac{1}{4 \pi \times 8.854 \times 10^{-12}} \approx 8.9875 \times 10^9 \, Nm^2/C^2 \][/tex]
- The charge [tex]\( q = 0.009 \times 10^{-6} \, C \)[/tex].
- Distance [tex]\( r = 3 \, m \)[/tex].
Now plug these into the formula:
[tex]\[ E = \frac{8.9875 \times 10^9 \, Nm^2/C^2 \times 0.009 \times 10^{-6} \, C}{3^2 \, m^2} \][/tex]
[tex]\[ E = \frac{8.9875 \times 10^9 \times 0.009 \times 10^{-6}}{9} \][/tex]
[tex]\[ E = \frac{8.9875 \times 0.009 \times 10^3}{9} \][/tex]
[tex]\[ E \approx 8.987742437988214 \, N/C \][/tex]
### Conclusion
The distance from the origin to point [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex] is [tex]\( 3 \, m \)[/tex], and the intensity of the electric field at that point due to the given charge is approximately [tex]\( 8.987742437988214 \, N/C \)[/tex].
### Given Data
1. Charge (q): The charge is [tex]\(0.009 \mu C\)[/tex].
- Convert microcoulombs to coulombs: [tex]\(0.009 \mu C = 0.009 \times 10^{-6} C = 0.009 \times 10^{-6} C\)[/tex].
2. Permittivity of Free Space (ε₀): The permittivity of free space is [tex]\(8.854 \times 10^{-12} \, F/m\)[/tex].
3. Coordinates (x, y, z): The point at which we need to find the electric field intensity is [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex].
### Step-by-Step Solution
#### Step 1: Calculate the Distance from the Origin
- The charge is placed at the origin, [tex]\((0, 0, 0) \, m\)[/tex].
- The point at which we need to calculate the electric field is [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex].
- The distance [tex]\(r\)[/tex] from the origin to the point is calculated using the distance formula:
[tex]\[ r = \sqrt{(\sqrt{2} - 0)^2 + (\sqrt{7} - 0)^2 + (0 - 0)^2} \][/tex]
Simplifying inside the square root:
[tex]\[ r = \sqrt{(\sqrt{2})^2 + (\sqrt{7})^2} \][/tex]
[tex]\[ r = \sqrt{2 + 7} \][/tex]
[tex]\[ r = \sqrt{9} \][/tex]
[tex]\[ r = 3 \, m \][/tex]
#### Step 2: Electric Field Intensity Calculation
- The electric field intensity [tex]\(E\)[/tex] due to a point charge is given by Coulomb's Law:
[tex]\[ E = \frac{k |q|}{r^2} \][/tex]
where [tex]\( k \)[/tex] (constant) can be expressed as [tex]\( \frac{1}{4 \pi \epsilon_0} \)[/tex], and [tex]\(\epsilon_0\)[/tex] is the permittivity of free space.
Substituting the known values:
[tex]\[ k = \frac{1}{4 \pi \epsilon_0} = \frac{1}{4 \pi \times 8.854 \times 10^{-12}} \approx 8.9875 \times 10^9 \, Nm^2/C^2 \][/tex]
- The charge [tex]\( q = 0.009 \times 10^{-6} \, C \)[/tex].
- Distance [tex]\( r = 3 \, m \)[/tex].
Now plug these into the formula:
[tex]\[ E = \frac{8.9875 \times 10^9 \, Nm^2/C^2 \times 0.009 \times 10^{-6} \, C}{3^2 \, m^2} \][/tex]
[tex]\[ E = \frac{8.9875 \times 10^9 \times 0.009 \times 10^{-6}}{9} \][/tex]
[tex]\[ E = \frac{8.9875 \times 0.009 \times 10^3}{9} \][/tex]
[tex]\[ E \approx 8.987742437988214 \, N/C \][/tex]
### Conclusion
The distance from the origin to point [tex]\((\sqrt{2}, \sqrt{7}, 0) \, m\)[/tex] is [tex]\( 3 \, m \)[/tex], and the intensity of the electric field at that point due to the given charge is approximately [tex]\( 8.987742437988214 \, N/C \)[/tex].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
