An infinite sheet with a surface charge density of -2.1 micro-coulombs/m2 runs parallel to the x axis. Another infinite sheet a surface charge density of +2.1 micro-coulombs/m2 runs parallel to the y axis. A charge of -4.1 micro-coulombs moves at a 45 degree angle towards the negative sheet as shown in the sketch. If the initial velocity of the charge was 10.7 m/s and it's mass is 0.22 kg, how far does it travel before it stops in meters? Ignore the weight force.
To solve this problem, we can use the principle of conservation of energy. The initial kinetic energy of the charge is equal to the work done by the electric field to bring it to a stop.
The initial kinetic energy (KE_initial) of the charge is given by:
KE_initial = (1/2) * m * v^2
where m is the mass of the charge and v is its initial velocity.
The work done by the electric field (W_electric) is given by:
W_electric = q * (ΔV)
where q is the charge and (ΔV) is the change in electric potential.
Since the charge is moving at a 45-degree angle towards the negative sheet, only the electric field from the negative sheet affects its motion. The electric potential (V) due to an infinite sheet with surface charge density σ is given by:
V = σ / (2 * ε_0)
where ε_0 is the electric constant.
Since the surface charge density of the negative sheet is -2.1 micro-coulombs/m^2 and the charge of the moving charge is -4.1 micro-coulombs, the electric potential (V_negative) due to the negative sheet is:
V_negative = (-2.1 * 10^-6 C/m^2) / (2 * ε_0)
The change in electric potential (ΔV) is the difference in electric potential between the initial and final positions of the charge:
ΔV = V_initial - V_final
To bring the charge to a stop, the final electric potential (V_final) would be zero.
Therefore, the work done by the electric field is:
W_electric = q * (ΔV)
= q * (V_initial - V_final)
= q * (V_initial - 0)
= q * V_initial
Substituting the values into the equation:
W_electric = (-4.1 * 10^-6 C) * V_negative
The work done by the electric field is equal to the initial kinetic energy (KE_initial) of the charge:
W_electric = KE_initial
Setting the two equations equal to each other and solving for the distance traveled (d):
(-4.1 * 10^-6 C) * V_negative = (1/2) * (0.22 kg) * (10.7 m/s)^2
Simplifying the equation, we can solve for d:
d = [(-4.1 * 10^-6 C) * V_negative] / [(1/2) * (0.22 kg) * (10.7 m/s)^2]
Substituting the values and calculating the expression will give you the distance traveled before the charge stops.