Question

Two laser beams with wavelengths λ1=596 nm and λ;2=574 nm are aimed at the same point. The electric field from each laser in the y-direction behaves as the function, Ei,y(x,t)=A(sin2πfit−2πxλi) for i=1 or 2 , and they propagate at a speed of light c . Both fields have the same amplitude, A=1 N/C . Find the value of the total electric field in the y-direction Ey�y, in newtons per coulomb, at a time of exactly one femtosecond (10−1510−15 seconds) if the position the lasers is aimed at is exactly 100100 nanometers away from each laser.

Answer 1

The total electric field in the y-direction, Ey, at a point 100 nanometers away from two laser pointers at t = 1 femtosecond, is calculated by summing the individual fields generated by each laser. This is done by substituting the given time and position into the electric field equations for each laser and using the property that their amplitudes are the same and the waves are in phase.

Step-by-step explanation:

To calculate the total electric field in the y-direction, Ey, at a given point 100 nanometers away from two laser pointers at time t = 1 femtosecond, we must consider the superposition principle, which states that the resultant field at a point is the vector sum of fields produced by each source independently.

The given electric field equations for each laser beam are E1,y(x,t) = A(sin(2πf1t - 2πx/λ1)) and E2,y(x,t) = A(sin(2πf2t - 2πx/λ2)), both with wavelengths λ1 = 596 nm and λ2 = 574 nm, respectively, and amplitude A=1 N/C. Since both sources are in phase and at the same amplitude, their effects simply add at the point of interest.

The frequencies f1 and f2 can be found using the relationship f = c/λ, where c is the speed of light. To find the total field at t = 1 fs (1 x 10^-15 s) and x = 100 nm (1 x 10^-7 m), we must substitute these values into the electric field equations and add the resulting fields.

Answer 2

The total electric field in the y-direction at a time of one femtosecond can be found by adding the electric fields from each laser beam. At a position 100 nanometers away from each laser, the total electric field is 0.3676 N/C.

Step-by-step explanation:

The total electric field Ey,y in the y-direction can be calculated by adding the electric fields from each laser beam at a given time. Given that the position the lasers are aimed at is 100 nanometers away from each laser and the time is one femtosecond (10^-15 seconds), we can use the formula Ei,y(x,t) = A(sin(2πfit) - 2πx/λi) to find the electric field from each laser beam at that position and time. For λ1 = 596 nm, the value of E1,y at the specified position and time is E1,y(x,t) = 1(sin(2π(3 × 10^8 Hz)(10^-15 s)) - 2π(100 × 10^-9 m)/(596 × 10^-9 m)) = 1N/C. For λ2 = 574 nm, the value of E2,y at the specified position and time is E2,y(x,t) = 1(sin(2π(3 × 10^8 Hz)(10^-15 s)) - 2π(100 × 10^-9 m)/(574 × 10^-9 m)) = -0.6324N/C Adding these two electric fields gives the total electric field Ey,y = E1,y + E2,y = 1N/C + (-0.6324N/C) = 0.3676N/C.