As a proton passes the origin, its velocity is $3.0\times {10}^5\mathrm{m}/\mathrm{s}$in the positive x direction. What is the magnitude of the magnetic field at the point ($12.0\mathrm{cm}$, $5.0\mathrm{cm}$, $0.0\mathrm{cm}$)?

• A. $1.2\times {10}^{-20}\mathrm{T}$
• B. $5.2\times {10}^{-20}\mathrm{T}$
• C. $9.2\times {10}^{-20}\mathrm{T}$
• D. $1.8\times {10}^{-19}\mathrm{T}$
• E. $2.4\times {10}^{-19}\mathrm{T}$
• F. 1.1 x 10^-19 T

Answer: (F)

Verified step by step guidance

1. Identify the relevant formula for the magnetic field due to a moving charge, which is given by the Biot-Savart Law: \( \mathbf{B} = \frac{\mu_0}{4\pi} \frac{q \mathbf{v} \times \mathbf{r}}{r^3} \), where \( \mathbf{B} \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( q \) is the charge, \( \mathbf{v} \) is the velocity, and \( \mathbf{r} \) is the position vector from the charge to the point of interest.
2. Convert the position coordinates from centimeters to meters for consistency in SI units: \( (12.0\, \text{cm}, 5.0\, \text{cm}, 0.0\, \text{cm}) \) becomes \( (0.12\, \text{m}, 0.05\, \text{m}, 0.0\, \text{m}) \).
3. Calculate the position vector \( \mathbf{r} \) from the origin to the point of interest: \( \mathbf{r} = (0.12\, \text{m}, 0.05\, \text{m}, 0.0\, \text{m}) \). The magnitude \( r \) is given by \( r = \sqrt{(0.12)^2 + (0.05)^2 + (0.0)^2} \).
4. Determine the cross product \( \mathbf{v} \times \mathbf{r} \). Since \( \mathbf{v} = (3.0 \times 10^5\, \text{m/s}, 0, 0) \) and \( \mathbf{r} = (0.12, 0.05, 0) \), the cross product is calculated using the determinant: \( \mathbf{v} \times \mathbf{r} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3.0 \times 10^5 & 0 & 0 \\ 0.12 & 0.05 & 0 \end{vmatrix} \).
5. Substitute the values into the Biot-Savart Law to find the magnitude of the magnetic field \( B \): \( B = \frac{\mu_0}{4\pi} \frac{q \| \mathbf{v} \times \mathbf{r} \|}{r^3} \). Use the known values for \( \mu_0 \), the charge of a proton \( q = 1.6 \times 10^{-19}\, \text{C} \), and the calculated \( \| \mathbf{v} \times \mathbf{r} \| \) and \( r \).