Calculation of Wavelength and Frequency:
Given: By = 2 × 10–7 sin (0.5 × 103x - 1.5 × 1011t) T

The equation for a plane wave in one dimension can be written as:
B(x, t) = B0 sin(kx - ωt + ϕ)

Comparing the given equation with the general equation, we can equate the terms to find the values of k and ω.

Comparing the x terms:
0.5 × 103x = kx
k = 0.5 × 103 rad/m

Comparing the t terms:
1.5 × 1011t = ωt
ω = 1.5 × 1011 rad/s

Wavelength (λ) can be calculated using the formula:
λ = 2π/k

Plugging in the value of k, we get:
λ = 2π/(0.5 × 103) = 4π × 103 m = 12.57 km

Frequency (f) can be calculated using the formula:
f = ω/2π

Plugging in the value of ω, we get:
f = (1.5 × 1011)/(2π) ≈ 23.84 GHz

Equation of the Electric Field:
In an electromagnetic wave, the electric field (E) and magnetic field (B) are related by the equation:
c = E/B

where c is the speed of light in a vacuum.

The speed of light in a vacuum is approximately 3 × 108 m/s.

Rearranging the equation, we can solve for E:
E = cB

Plugging in the values, we get:
E = (3 × 108 m/s)(2 × 10–7 sin (0.5 × 103x - 1.5 × 1011t)) = 6 × 101 m/s sin (0.5 × 103x - 1.5 × 1011t) V/m

Thus, the equation of the electric field is E = 6 × 101 m/s sin (0.5 × 103x - 1.5 × 1011t) V/m.

Explanation:
- The wavelength (λ) represents the spatial period of the wave and is the distance between two consecutive peaks or troughs.
- The frequency (f) represents the temporal period of the wave and is the number of complete cycles per second.
- The equation of the electric field (E) is derived from the relationship between the electric and magnetic fields in an electromagnetic wave.
- By comparing the given equation with the general equation for a plane wave, we can determine the values of k and ω, which are used to calculate the wavelength and frequency.
- The speed of light in a vacuum (c) is used to relate the electric and magnetic fields in the wave.