Picture the following:
A square with 4 point charges. Charges Q are diagonal from eachother, and charges -Q are diagonal from eachother. The length of each side is "a".
One of the -Q charges is "kicked" away with an outward initial speed of V0, while the three other charges are held at rest. If the moving charge has a mass m, what is its speed when it is infinitely far from the other charges?
Express the answer in terms of Q, m, V0, a and appropriate constants.
To find the speed of the moving charge when it is infinitely far from the other charges, we can use the principle of conservation of mechanical energy. In this case, we can consider the net initial mechanical energy of the system (consisting of the moving charge and the other three charges) to be zero, since the three charges are held at rest initially.
The net mechanical energy of a system is given by the sum of its kinetic energy (KE) and potential energy (PE). Therefore, the initial energy of the system is:
E_initial = KE_initial + PE_initial
Since the three charges are held at rest, they do not contribute to the initial kinetic energy. Thus, the initial kinetic energy (KE_initial) is zero.
The initial potential energy (PE_initial) is the sum of the electric potential energy between the moving charge and the other three charges. The electric potential energy of two charges Q and -Q separated by a distance r is given by the formula:
PE = k * (Q * (-Q)) / r , where k is the Coulomb constant.
In this case, we have two pairs of charges with equal magnitudes (Q and -Q), and their distances from each other are equal (a). So, the initial potential energy between the moving charge and the other three charges is:
PE_initial = (k * (Q * (-Q))) / a + (k * (Q * (-Q))) / a
Simplifying this expression:
PE_initial = -2k * Q^2 / a
Now let's consider the final state where the moving charge is infinitely far from the other charges. At that point, all the charges are so far apart that their interaction can be considered negligible. Therefore, the final potential energy (PE_final) is zero.
E_final = KE_final + PE_final
Since the final potential energy is zero:
E_final = KE_final + 0
Now, to find the final speed of the moving charge (v_final), we need to equate the initial and final energies:
E_initial = E_final
KE_initial + PE_initial = KE_final
Since KE_initial is zero, we can simplify the equation:
PE_initial = KE_final
-2k * Q^2 / a = (1/2) * m * v_final^2
Now, we can solve for v_final:
v_final = sqrt((-2k * Q^2 / a) * (2 / m)))
The final speed of the moving charge when it is infinitely far from the other charges is given by the above equation.