A π- ("pi-minus") particle, which has charge -e, is at location ‹ 3.00e10-9, -2.00e10-9, -3.00e10-9› m. What is the electric field at location < -3.00e10-9, 3.00e10-9, 4.00e10-9> m, due to the π- particle?
To calculate the electric field at a point due to a π- particle, we can use Coulomb's law. Coulomb's law states that the electric field produced by a charged particle at a given point is given by:
E = k * (q / r^2)
Where:
- E is the electric field
- k is the Coulomb's constant (k = 8.99e9 N m^2/C^2)
- q is the charge on the particle
- r is the distance between the particle and the point
Given:
- Charge on the π- particle, q = -e
- Position of the π- particle, r1 = <3.00e-9, -2.00e-9, -3.00e-9> m
- Position of the point where we need to find the electric field, r2 = <-3.00e-9, 3.00e-9, 4.00e-9> m
Let's calculate the electric field step by step:
Step 1: Calculate the distance between the π- particle and the point.
r = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
= sqrt((-3.00e-9 - 3.00e-9)^2 + (3.00e-9 - (-2.00e-9))^2 + (4.00e-9 - (-3.00e-9))^2)
Step 2: Substitute the values into the equation for the electric field.
E = k * (q / r^2)
= 8.99e9 * (-e) / r^2
Now, let's calculate the value of the electric field.
To determine the electric field at a specific location due to a charged particle, you need to calculate the electric field vector using Coulomb's Law and vector addition. Here's how you can do it step by step:
1. Determine the distance between the charged particle and the point where you want to find the electric field. In this case, the distance is the difference between the two locations provided:
Distance = (∆x, ∆y, ∆z) = (< -3.00e10-9, 3.00e10-9, 4.00e10-9> m) - (‹ 3.00e10-9, -2.00e10-9, -3.00e10-9› m)
Distance = < -3.00e10-9 - 3.00e10-9, 3.00e10-9 - (-2.00e10-9), 4.00e10-9 - (-3.00e10-9) > m
2. Calculate the magnitude of the distance vector:
|Distance| = √(∆x² + ∆y² + ∆z²)
3. Calculate the electric field magnitude using Coulomb's Law:
Electric field magnitude = k * |Charge| / |Distance|²
Where:
- k is the electrostatic constant (k = 8.99 * 10^9 Nm²/C²)
- |Charge| is the absolute value of the charge of the π- particle (|Charge| = |-e| = e, where e is the elementary charge, e = 1.6 * 10^-19 C)
- |Distance| is the magnitude of the distance vector calculated in step 2.
4. Calculate the electric field vector direction:
The electric field vector points away from a positive charge and towards a negative charge along the line connecting the charges. So, the direction of the electric field vector is the same as the direction of the distance vector.
5. Convert the electric field magnitude and direction to vector form:
Electric field vector = Electric field magnitude * Distance / |Distance|
Substitute the values into the equations and calculate to find the electric field at the given location.