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A particle with a charge of -1.24 × 108 C is moving with instantaneous velocity v= 14.19 × 104 m/s)ı^ + (-3.85 × 104 m/s)ȷ^. What is the force exerted on this particle by a magnetic field (a) B= (1.40 T)ı^ and (b) B= (1.40 T) k^ ?

Short Answer

Expert verified
(a) F1=0ı^+0ȷ^6.68×104k^ N; (b) F2=6.68×104ı^+2.47×103ȷ^ N.

Step by step solution

01

Understand the Given Problem

We need to find the force exerted by a magnetic field on a charge with a specific velocity vector in two different scenarios. The charge is given as q=1.24×108 C, and the velocity vector is v=14.19×104ı^3.85×104ȷ^ m/s. We need to find the force for two different magnetic fields: B1=1.40ı^ T and B2=1.40k^ T.
02

Use the Lorentz Force Formula

The magnetic force F on a charged particle moving with velocity v in a magnetic field B is given by the cross product F=q(v×B). We'll need to compute this for each magnetic field scenario.
03

Calculate Cross Product for (a) B=1.40ı^

For B1=1.40ı^, the cross product is v×B1=(14.19×104ı^3.85×104ȷ^)×1.40ı^. Since a vector crossed with itself is zero, and the cross product between vectors on the same axis is zero, this simplifies to: v×B1=0ı^+0ȷ^+(1.40×(3.85×104))k^. Calculating gives: v×B1=0ı^+0ȷ^+5.39×104k^.
04

Calculate Force for (a)

Using q=1.24×108 C and the cross product from Step 3, we find: F1=1.24×108×(0ı^+0ȷ^+5.39×104k^). This gives: F1=0ı^+0ȷ^6.68×104k^ N.
05

Calculate Cross Product for (b) B=1.40k^

For B2=1.40k^, the cross product is v×B2=(14.19×104ı^3.85×104ȷ^)×1.40k^.The result is calculated using the determinant method for cross products:(3.85×104×1.40)ı^(14.19×104×1.40)ȷ^. Which results in:v×B2=5.39×104ı^1.99×105ȷ^+0k^.
06

Calculate Force for (b)

Using q=1.24×108 C and the cross product from Step 5, we find:F2=1.24×108(5.39×104ı^1.99×105ȷ^).This results in: F2=6.68×104ı^+2.47×103ȷ^ N.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It's visualized as "lines of force" that exit through a magnet's north pole and enter through its south pole. Magnetic fields can be created by permanent magnets or flows of electric current. When a moving charge enters a magnetic field, it experiences a force. This force's direction depends on the charge's velocity and the magnetic field's direction. The strength of a magnetic field is measured in Tesla (T).

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.
In the context of the original exercise, recognizing which direction the magnetic fields (B1 and B2) are acting in is crucial for calculating the resulting force on the charged particle.
Cross Product
The cross product is a vector multiplication operation that yields a vector perpendicular to two input vectors in three-dimensional space. This operation is central in physics, particularly in electromagnetism, where it is used to determine the force experienced by a charged particle within a magnetic field. Mathematically, the cross product of two vectors a and b is denoted as a×b. The magnitude of the resulting vector is given by |a||b|sinθ, where θ is the angle between the original vectors.
Key Characteristics:
  • A cross product of parallel vectors is zero since sin(0)=0.
  • The resulting vector is perpendicular to the original vectors.
  • It is used to calculate torques, rotations, and forces like the Lorentz force.
In the exercise, the cross product helps determine v×B, which then allows calculation of the Lorentz force on the particle through F=q(v×B).
Vector Calculus
Vector calculus is a mathematical tool used to study vector fields, which are functions assigning a vector to each point in space. It's vital in physics for understanding phenomena involving fields, such as electromagnetic fields. Vector operations, such as addition, scalar multiplication, and cross products, help solve complex physical problems.

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.
In our task, vector calculus is applied to the velocity vector v and the magnetic field B to determine the force using the cross product. This application exemplifies the real-world utility of vector calculus in physics.

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Most popular questions from this chapter

A singly charged ion of 7Li (an isotope of lithium) has a mass of 1.16 × 1026 kg. It is accelerated through a potential difference of 220 V and then enters a magnetic field with magnitude 0.874 T perpendicular to the path of the ion. What is the radius of the ion's path in the magnetic field?

A 150-g ball containing 4.00 × 108 excess electrons is dropped into a 125-m vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal magnetic field that has magnitude 0.250 T and direction from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

A particle with initial velocity v$$0= (5.85 × 103m/s)ȷ^ enters a region of uniform electric and magnetic fields. The magnetic field in the region is B= - (1.35 T)k^. Calculate the magnitude and direction of the electric field in the region if the particle is to pass through undeflected, for a particle of charge (a) +0.640 nC and (b) -0.320 nC. You can ignore the weight of the particle.

A plastic circular loop has radius R, and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then rotated around its central axis, perpendicular to the plane of the loop, with angular speed ω. If the loop is in a region where there is a uniform magnetic field B directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of 6.64 × 1027 kg) traveling horizontally at 35.6 km>s enters a uniform, vertical, 1.80-T magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

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