dF 
y
​
 = 
r 
2
 
k 
e
​
 λqdy
​
 = 
[0.02 
2
 +(0.02−y) 
2
 ]
(8.99×10 
9
 )(8×10 
−9
 )(0.4×10 
−9
 )dy
​
 


It seems you're working with an equation involving differential elements \( dF_y \) and an integration related to force in a specific direction (possibly electric force) or some other physical quantity depending on the parameters. Let's break down what you have:

The equation provided is:

\[
dF_y = r^2 ke \lambda q \, dy = \left[0.02^2 + (0.02 - y)^2\right] (8.99 \times 10^9) (8 \times 10^{-9}) (0.4 \times 10^{-9}) \, dy
\]

Where:
- \( dF_y \) might represent the differential force in the \( y \)-direction,
- \( r^2 \), \( k \), and \( e \) are constants,
- \( \lambda \) and \( q \) could refer to a charge distribution or some other quantities involving charge.

### Step-by-step analysis:

1. **Constant terms**: 
   - \( r^2 = 0.02^2 + (0.02 - y)^2 \)
   - \( k = 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \) (Coulomb's constant)
   - \( \lambda = 8 \times 10^{-9} \, \text{C/m} \) (linear charge density)
   - \( q = 0.4 \times 10^{-9} \, \text{C} \) (charge)

2. **Simplifying the expression**:
   \[
   dF_y = \left[0.02^2 + (0.02 - y)^2\right] (8.99 \times 10^9) (8 \times 10^{-9}) (0.4 \times 10^{-9}) \, dy
   \]

   Compute the constant term first:
   \[
   (8.99 \times 10^9) (8 \times 10^{-9}) (0.4 \times 10^{-9}) = 2.8768 \times 10^{-8}
   \]

   So now the equation becomes:
   \[
   dF_y = \left[0.02^2 + (0.02 - y)^2\right] \cdot 2.8768 \times 10^{-8} \, dy
   \]

3. **Next step**: To find the total force in the \( y \)-direction, you'd likely integrate this expression over a specific range of \( y \)-values.

Do you want to go through the integration process or discuss a particular step? Let me know!

---

### Related Questions:
1. What physical system does this equation describe (electric fields, forces, etc.)?
2. How are the limits of integration determined for this equation?
3. How does the variable \( y \) affect the result of the integral in this system?
4. Could this equation represent the interaction between two charges? If so, how would the distance between them modify the result?
5. How would the solution change if different values for charge or distance were used?

**Tip**: Whenever working with differential equations involving forces, ensure that the limits of integration are clearly defined to calculate the total effect accurately.

dFy = r^2 ke λq dy = [0.02^2 + (0.02−y)^2](8.99×10^9)(8×10^−9)(0.4×10^−9) dy

Understand how to calculate differential force in the y-direction using integration, involving key concepts in electrostatics, such as Coulomb's law and differential calculus. This detailed explanation includes charge distribution, constants, and step-by-step integration suitable for Grades 11-12.

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