Chapter 27: Problem 1
A particle with a charge of -1.24 \(\times\) 10\(^{-8}\) C is moving with instantaneous velocity \(\vec{v} =\) 14.19 \(\times\) 10\(^4\) m/s)\(\hat{\imath}\) + (-3.85 \(\times\) 10\(^4\) m/s)\(\hat{\jmath}\). What is the force exerted on this particle by a magnetic field (a) \(\overrightarrow{B} =\) (1.40 T)\(\hat{\imath}\) and (b) \(\overrightarrow{B} =\) (1.40 T) \(\hat{k}\) ?
Short Answer
Step by step solution
Understand the Given Problem
Use the Lorentz Force Formula
Calculate Cross Product for (a) \( \overrightarrow{B} = 1.40 \hat{\imath} \)
Calculate Force for (a)
Calculate Cross Product for (b) \( \overrightarrow{B} = 1.40 \hat{k} \)
Calculate Force for (b)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Magnetic Field
Key Points:
- A magnetic field exerts a force on moving charges.
- The magnetic field's direction and the charge’s motion will determine the direction of this force.
- Strong magnetic fields exert stronger forces on charges.
Cross Product
Key Characteristics:
- A cross product of parallel vectors is zero since \( \sin(0^\circ) = 0 \).
- The resulting vector is perpendicular to the original vectors.
- It is used to calculate torques, rotations, and forces like the Lorentz force.
Vector Calculus
Important Operations:
- Vector Addition: Combines two vectors' magnitudes and directions.
- Scalar Multiplication: Involves multiplying a vector by a scalar (number), scaling its magnitude.
- Cross Product: Yields a vector perpendicular to two given vectors, especially crucial in calculating forces in electromagnetism.