To determine the distance \( L \) between two charges, \( +Q \) and \( -q \), such that they remain stationary in an electric field \( E \) directed from the negative to the positive charge, we need to analyze the forces acting on each charge. This involves understanding the balance of electric and gravitational forces.
Understanding the Forces at Play
When the charges are placed in an electric field, they experience forces due to the field as well as gravitational forces due to their masses. Let's break down the forces acting on each charge:
- Force on Charge +Q: The force \( F_Q \) acting on charge \( +Q \) due to the electric field \( E \) is given by:
F_Q = Q \cdot E
- Force on Charge -q: The force \( F_q \) acting on charge \( -q \) is:
F_q = -q \cdot E
Since the electric field is directed from the negative charge to the positive charge, the force on \( -q \) will be in the opposite direction of the electric field, which means it will also be attracted toward \( +Q \).
Setting Up the Equations
Next, we need to consider the gravitational forces acting on both charges:
- Gravitational Force on +Q: This force \( F_{gQ} \) is given by:
F_{gQ} = m \cdot g
- Gravitational Force on -q: Similarly, the gravitational force \( F_{gq} \) is:
F_{gq} = M \cdot g
Condition for Equilibrium
For the system to remain constant (in equilibrium), the net force acting on each charge must be zero. This means that the electric force must balance the gravitational force for both charges:
- For charge \( +Q \):
Q \cdot E - m \cdot g = 0
- For charge \( -q \):
-q \cdot E - M \cdot g = 0
Solving for the Distance L
From the equations above, we can express the electric field \( E \) in terms of the forces:
- From \( +Q \):
E = \frac{m \cdot g}{Q}
- From \( -q \):
E = \frac{M \cdot g}{q}
Setting these two expressions for \( E \) equal to each other gives us:
\(\frac{m \cdot g}{Q} = \frac{M \cdot g}{q}\)
By simplifying this equation, we can find a relationship between the masses and charges:
\( \frac{m}{Q} = \frac{M}{q} \)
Now, if we want to maintain a constant distance \( L \) between the charges while they are in the electric field, we can express \( L \) in terms of the charges and masses. The exact relationship will depend on the specific configuration and the values of \( Q \), \( q \), \( m \), and \( M \). However, the key takeaway is that the distance \( L \) must be such that the forces due to the electric field and gravitational forces are balanced for both charges.
Final Thoughts
In summary, to keep the charges stationary in the electric field, the distance \( L \) must be adjusted based on the ratio of the masses and charges. This ensures that the forces acting on each charge are equal and opposite, allowing them to maintain their positions without moving toward or away from each other.
