A triangle with equal sides of length 10 cm has -1.5 nC charged objects at each corner. Determine the magnitude of the electrical force exerted on the object at the top corner due... A triangle with equal sides of length 10 cm has -1.5 nC charged objects at each corner. Determine the magnitude of the electrical force exerted on the object at the top corner due to the two objects at the base of the triangle.
Understand the Problem
The question is asking us to calculate the magnitude of the electrical force exerted on one charged object located at the top corner of an equilateral triangle by the two charged objects at the base. This involves applying Coulomb's law and vector addition to find the resultant force.
Answer
The magnitude of the electrical force is given by $$ F_{net} = \sqrt{\left( F_{23} \cdot \sin(60^\circ) \right)^2 + \left( -F_{13} + F_{23} \cdot \cos(60^\circ) \right)^2 } $$
Answer for screen readers
The magnitude of the electrical force exerted on charge ( q_3 ) is given by the calculation described above, which results in $$ F_{net} = \sqrt{\left( F_{23} \cdot \sin(60^\circ) \right)^2 + \left( -F_{13} + F_{23} \cdot \cos(60^\circ) \right)^2 } $$
Steps to Solve
- Identify charges and distances
Let's denote the charges as ( q_1 ) and ( q_2 ) at the base of the triangle, and ( q_3 ) at the top corner. Let the side length of the equilateral triangle be ( a ). The distance from ( q_3 ) to each of ( q_1 ) and ( q_2 ) is ( a ).
- Apply Coulomb's Law
Using Coulomb's law, the electrical force between two charges is given by: $$ F = k \frac{|q_1 q_2|}{r^2} $$ where ( k ) is Coulomb's constant, ( r ) is the distance between the charges.
- Calculate individual forces
Calculate the force exerted on ( q_3 ) by ( q_1 ) and ( q_2 ):
- The force due to ( q_1 ) is: $$ F_{13} = k \frac{|q_3 q_1|}{a^2} $$
- The force due to ( q_2 ) is: $$ F_{23} = k \frac{|q_3 q_2|}{a^2} $$
- Determine direction of forces
Since the forces act in different directions:
- ( F_{13} ) acts straight down towards ( q_1 ).
- ( F_{23} ) makes an angle of ( 60^\circ ) with the vertical line from ( q_3 ) to the midpoint between ( q_1 ) and ( q_2 ).
- Resolve forces into components
-
Resolve ( F_{13} ) (downward):
- ( F_{13x} = 0 )
- ( F_{13y} = -F_{13} )
-
Resolve ( F_{23} ):
- ( F_{23x} = F_{23} \cdot \sin(60^\circ) )
- ( F_{23y} = -F_{23} \cdot \cos(60^\circ) )
- Net force in each direction
Add up the forces in the x and y directions to get the net force:
- Net x-component: $$ F_{net_x} = F_{23x} + F_{13x} $$
- Net y-component: $$ F_{net_y} = F_{13y} + F_{23y} $$
- Calculate magnitude of the resultant force
The resultant force ( F_{net} ) can be calculated using the Pythagorean theorem: $$ F_{net} = \sqrt{F_{net_x}^2 + F_{net_y}^2} $$
The magnitude of the electrical force exerted on charge ( q_3 ) is given by the calculation described above, which results in $$ F_{net} = \sqrt{\left( F_{23} \cdot \sin(60^\circ) \right)^2 + \left( -F_{13} + F_{23} \cdot \cos(60^\circ) \right)^2 } $$
More Information
This problem demonstrates the relationship between electrical charges and the forces they exert on each other according to Coulomb's law. It requires knowledge of both vector addition and basic trigonometry to resolve the forces into components.
Tips
- Neglecting to resolve the forces into their x and y components can lead to incorrect resultant force calculations.
- Miscalculating the angles involved in the vector forces is another common mistake; ensure to visualize the angles in the triangle properly.
AI-generated content may contain errors. Please verify critical information
