Question

A proton moves along the x-axis with vₓ = 1.0 x 10⁷ m/s. As it passes the origin, what are the strength and direction of the magnetic field at the (x, y, z) positions (a) (1 cm, 0 cm, 0 cm), (b) (0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm)?

Accepted Answer

Step 1: Recall the formula for the magnetic field produced by a moving charge. The magnetic field at a point due to a moving charge is given by **B = (μ₀ / 4π) * (q * v × r̂) / r²**, where μ₀ is the permeability of free space, q is the charge, v is the velocity vector, r̂ is the unit vector pointing from the charge to the observation point, and r is the distance between the charge and the observation point. Step 2: Identify the given values. The charge of a proton is q = 1.6 × 10⁻¹⁹ C, the velocity vector is **v = (1.0 × 10⁷ m/s) î**, and the observation points are (a) (1 cm, 0 cm, 0 cm), (b) (0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm). Convert the distances to meters: 1 cm = 0.01 m, 2 cm = 0.02 m. Step 3: For each observation point, calculate the distance vector **r** and the unit vector **r̂**. For example, for point (a), **r = (0.01 m) î**, and **r̂ = r / |r| = î**. Similarly, calculate **r** and **r̂** for points (b) and (c). Step 4: Compute the cross product **v × r̂** for each observation point. For point (a), since **v** and **r̂** are parallel, the cross product is zero, meaning the magnetic field at (1 cm, 0 cm, 0 cm) is zero. For points (b) and (c), use the right-hand rule to determine the direction of **v × r̂** and calculate its magnitude. Step 5: Substitute the values into the formula for **B** to find the magnitude and direction of the magnetic field at each point. Use the calculated **v × r̂**, the distance **r**, and the constants μ₀ = 4π × 10⁻⁷ T·m/A and q = 1.6 × 10⁻¹⁹ C to determine the magnetic field strength at points (b) and (c).

A proton moves along the x-axis with vₓ = 1.0 x 10⁷ m/s. As it passes the origin, what are the strength and direction of the magnetic field at the (x, y, z) positions (a) (1 cm, 0 cm, 0 cm), (b) (0 cm, 1 cm, 0 cm), and (c) (0 cm, -2 cm, 0 cm)?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Magnetic Field Due to a Moving Charge

Right-Hand Rule

Coordinate System and Position Vectors