what potential difference U is needed such that a point charge (q) between a parralel plate capacitor leaves it at the upper right corner. the capacitor is separated by a distance "d" and the length of each plate is "l"
To find the potential difference U needed for the charge q to leave the capacitor and reach the upper right corner, we can consider the electric field between the plates.
The electric field between the parallel plates of a capacitor is given by:
E = σ / ε₀
Where σ is the surface charge density on the plates and ε₀ is the permittivity of free space.
The surface charge density σ can be calculated using the formula:
σ = q / A
Where q is the charge and A is the area of either plate.
The area of one plate can be calculated as:
A = l * d
Where l is the length of each plate and d is the separation distance between the plates.
Now, we can substitute the expressions for σ and A into the formula for the electric field:
E = (q / A) / ε₀
= q / (A * ε₀)
= q / (l * d * ε₀)
Next, we can determine the force (F) exerted on the charge q by the electric field using the formula:
F = q * E
Substituting the expression for E:
F = q * (q / (l * d * ε₀))
= q² / (l * d * ε₀)
For the charge q to leave the capacitor, the force F must be greater than or equal to the gravitational force acting on the charge. Let's assume the charge q has a mass m.
The gravitational force acting on the charge is given by:
F_gravity = m * g
Where g is the acceleration due to gravity.
Setting F equal to F_gravity:
q² / (l * d * ε₀) = m * g
Solving for q:
q = sqrt((m * g * l * d * ε₀))
To find the potential difference U, we can use the formula:
U = q / C
Where C is the capacitance of the capacitor. The capacitance of a parallel plate capacitor is given by:
C = ε₀ * A / d
Substituting the expression for A:
C = ε₀ * (l * d) / d
C = ε₀ * l
Finally, we can substitute the expression for q and C into the formula for U:
U = (sqrt(m * g * l * d * ε₀)) / (ε₀ * l)
= sqrt(m * g * d)
Thus, the potential difference U needed for the charge q to leave the capacitor and reach the upper right corner is given by:
U = sqrt(m * g * d)
To calculate the potential difference (U) needed for a point charge (q) to leave a parallel plate capacitor at the upper right corner, you can consider the electric field (E) between the plates.
The electric field between the plates of a parallel plate capacitor is given by:
E = σ / ε₀
Where:
- E is the electric field in between the plates,
- σ is the surface charge density on the plates,
- ε₀ is the permittivity of free space (a constant value).
The electric field causes a force on the charge (q) that will move it upwards. This force is given by:
F = qE
To get the potential difference (U), you need to multiply the force (F) by the distance (d) over which the charge moves. Therefore, the potential difference (U) can be calculated as:
U = F * d / q
Now, since we have E = σ / ε₀ and F = qE, we can substitute these values into the equation for U:
U = (qE) * d / q
Now, canceling out the q, we get:
U = E * d
The electric field (E) between the plates of a parallel plate capacitor in terms of the surface charge density (σ) and the separation distance (d) can be expressed as:
E = σ / (ε₀ * d)
Substituting this expression for E into the equation for U, we get:
U = (σ / (ε₀ * d)) * d
Simplifying, we have:
U = σ / ε₀
In summary, the potential difference (U) needed for the point charge (q) to leave the capacitor at the upper right corner is equal to the surface charge density (σ) on the plates of the capacitor divided by the permittivity of free space (ε₀).