What is (Fnet3)x, the x-component of the net force exerted by the two charges on a third charge placed between them?

Steps to Solve

1. Identify the charges and their positions

Let the three charges be ( q_1, q_2, ) and ( q_3 ) with positions ( x_1, x_2, ) and ( x_3 ), respectively. Define ( q_1 ) at ( x_1 ), ( q_2 ) at ( x_2 ), and ( q_3 ) at ( x_3 ).

2. Calculate the force exerted by each charge on the third charge

Using Coulomb's Law, the force ( F ) between two charges is given by:

$$ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} $$

where ( k ) is Coulomb's constant and ( r ) is the distance between the charges.

3. Determine the distances

Calculate the distance ( r_{13} ) between charges ( q_1 ) and ( q_3 ):

$$ r_{13} = |x_3 - x_1| $$

Calculate the distance ( r_{23} ) between charges ( q_2 ) and ( q_3 ):

$$ r_{23} = |x_3 - x_2| $$

4. Calculate the individual forces

Calculate the force ( F_{13} ) from charge ( q_1 ) on charge ( q_3 ):

$$ F_{13} = k \cdot \frac{|q_1 \cdot q_3|}{r_{13}^2} $$

Determine the direction of ( F_{13} ) based on the charge signs.

Calculate the force ( F_{23} ) from charge ( q_2 ) on charge ( q_3 ):

$$ F_{23} = k \cdot \frac{|q_2 \cdot q_3|}{r_{23}^2} $$

Also determine the direction of ( F_{23} ) based on the charge signs.

5. Calculate the net force on charge ( q_3 )

Add the forces considering their directions:

$$ F_{\text{net}} = F_{13} + F_{23} $$

Make sure to account for whether each force is attractive or repulsive to get the correct sign.

6. Find the x-component of the net force

If the problem specifies directions, separate components accordingly. The x-component ( F_{x,\text{net}} ) may be derived from the net force, which is already in the x-direction.