What is the magnetic force on a proton that is moving at $3.8 \times 10^7 m / s$ up through a magnetic field that is 0.72 T and pointing toward you? The charge on a proton is $1.6 \times 10^{-19} C$. Use $F=q v \times B \sin \theta$.
A. $6.1 \times 10^{-12} N$ left
B. $6.1 \times 10^{-12} N$ right
C. $4.4 \times 10^{-12} N$ right
D. $4.4 \times 10^{-12} N$ left
Answer (1)
Calculate the magnitude of the magnetic force using the formula: F = q v B sin θ .
Substitute the given values: F = ( 1.6 × 1 0 − 19 C ) × ( 3.8 × 1 0 7 m / s ) × ( 0.72 T ) × 1 .
Calculate the magnitude: F = 4.4 × 1 0 − 12 N .
Determine the direction using the right-hand rule: the force is to the right. Therefore, the final answer is 4.4 × 1 0 − 12 N right .
Explanation
Problem Setup and Given Values We are given the charge of a proton q = 1.6 × 1 0 − 19 C , its velocity v = 3.8 × 1 0 7 m / s , and the magnetic field B = 0.72 T . The angle θ between the velocity and the magnetic field is 9 0 \tIcirc , so sin θ = 1 . We need to find the magnetic force F on the proton using the formula F = q v B sin θ .
Substituting Values into the Formula Now, we substitute the given values into the formula: F = ( 1.6 × 1 0 − 19 C ) × ( 3.8 × 1 0 7 m / s ) × ( 0.72 T ) × sin ( 9 0 \tIcirc ) Since sin ( 9 0 \tIcirc ) = 1 , the equation simplifies to: F = ( 1.6 × 1 0 − 19 ) × ( 3.8 × 1 0 7 ) × ( 0.72 ) × 1
Calculating the Magnitude of the Force Calculating the magnitude of the force: F = 1.6 × 3.8 × 0.72 × 1 0 − 19 + 7 F = 4.3776 × 1 0 − 12 N So, the magnitude of the magnetic force is approximately 4.4 × 1 0 − 12 N .
Determining the Direction of the Force To determine the direction of the force, we use the right-hand rule. Point your fingers in the direction of the proton's velocity (upwards), and curl them in the direction of the magnetic field (towards you). Your thumb will point in the direction of the force. In this case, the force is directed to the right.
Final Answer Therefore, the magnetic force on the proton is 4.4 × 1 0 − 12 N to the right.
Examples
Magnetic forces are crucial in many real-world applications, such as in electric motors, which convert electrical energy into mechanical energy using magnetic forces on current-carrying wires. Also, magnetic resonance imaging (MRI) uses strong magnetic fields to produce detailed images of the inside of the human body, relying on the principles of magnetic forces acting on atomic nuclei. Understanding magnetic forces also helps in designing particle accelerators used in scientific research to study the fundamental building blocks of matter.
