Question

The Stanford Linear Accelerator accelerated a beam consisting of \(2.0 \cdot 10^{14}\) electrons per second through a potential difference of \(2.0 \cdot 10^{10} \mathrm{~V}\) a) Calculate the current in the beam. b) Calculate the power of the beam. c) Calculate the effective ohmic resistance of the accelerator.

Short answer

Answer: The current in the beam is 3.2 x 10^-5 A, the power of the beam is 6.4 x 10^5 W, and the effective ohmic resistance of the accelerator is 6.25 x 10^14 Ω.

Step-by-step solution

(Step 1: Calculate the current in the beam)

To calculate the current in the beam, we will use the formula I = n * q * v, where I is the current, n is the number of electrons per second, q is the charge of an electron, and v is the velocity of the electrons. In this case, n is given, q is a known constant (approximately \(1.6 \cdot 10^{-19} \mathrm{C}\)), and we can rearrange the formula to solve for I. $$I = n * q$$ $$I = (2.0 \cdot 10^{14}\ \mathrm{electrons/s}) * (1.6 \cdot 10^{-19}\ \mathrm{C/electron})$$ $$I = 3.2 \cdot 10^{-5}\ \mathrm{A}$$ The current in the beam is \(3.2 \cdot 10^{-5} \mathrm{A}\).

(Step 2: Calculate the power of the beam)

To calculate the power of the beam, we will use the formula P = IV, where P is the power, I is the current, and V is the potential difference. In this case, I and V are given, so we can plug in the values and solve for P. $$P = IV$$ $$P = (3.2 \cdot 10^{-5}\ \mathrm{A}) * (2.0 \cdot 10^{10}\ \mathrm{V})$$ $$P = 6.4 \cdot 10^5\ \mathrm{W}$$ The power of the beam is \(6.4 \cdot 10^5 \mathrm{W}\).

(Step 3: Calculate the effective ohmic resistance of the accelerator)

To calculate the effective ohmic resistance of the accelerator, we can use the formula P = I^2 R, where P is the power, I is the current, and R is the resistance. In this case, P and I are known, so we can rearrange the formula to solve for R. $$R = \frac{P}{I^2}$$ $$R = \frac{6.4 \cdot 10^5\ \mathrm{W}}{(3.2 \cdot 10^{-5}\ \mathrm{A})^2}$$ $$R = 6.25 \cdot 10^{14}\ \mathrm{\Omega}$$ The effective ohmic resistance of the accelerator is \(6.25 \cdot 10^{14} \mathrm{\Omega}\).