Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of 1012W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0μm in diameter, with the pulse lasting for 4.0 ns with an average power of 2.0×1012W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in W/m2) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

a. 80 J/cell b. 1.02 × 10²¹ W/m² c. 8.77 × 10¹¹ V/m d. 2.92 × 10³ T

Explanation:

a. We know that energy E = Pt where P = power and t = time

The total energy delivered to all the 100 cells is E = average power × time

average power = 2 × 10¹² W and time = 4 ns = 4 × 10⁻⁹ s

E = 2 × 10¹² W × 4 × 10⁻⁹ s = 8 × 10³ J

The energy per cell E₁ = E/100 = 8 × 10³ J/100 = 80 J/cell

b. Intensity, I = P/A where P = power per cell = 2 × 10¹² W/100 = 2 × 10¹⁰ W and A = area = πr². Since the cell is modeled as a disk of diameter d = 5.0μm, r = d/2 = 5.0 μm/2 = 2.5 μm = 2.5 × 10⁻⁶ m

I = P/A = P/πr² = 2 × 10¹⁰ W/π(2.5 × 10⁻⁶ m)² = 1.02 × 10²¹ W/m²

c. The intensity I = E²/2cμ₀ where E = maximum value of electric field, c = speed of light = 3 × 10⁸ m/s and μ₀ = 4π × 10⁻⁷ H/m

E = √(I2cμ₀) = √(2 × 1.02 × 10²¹ W/m² × 3 × 10⁸ m/s × 4π × 10⁻⁷ H/m) = 8.77 × 10¹¹ V/m

The maximum magnetic field B is gotten from E/B = c

B = E/c = 8.77 × 10¹¹ V/m/3 × 10⁸ m/s  = 2.92 × 10³ T