Chapter 3: Q44P (page 59)

In the product $\vec{F} = q \vec{v} \times \vec{B}$ , take $q = 2, \vec{v} = 2.0 \overset{⏜}{i} + 4.0 \overset{⏜}{j} + 60 \overset{⏜}{k}$ ,and $\vec{F} = 4.0 \overset{⏜}{i} - 20 \overset{⏜}{j} + 12 \overset{⏜}{k}$ . What then is $\vec{B}$ in unit-vector notation if $B_{x} = B_{y}$ ?

Short Answer

Expert verified

If $B_{x} = B_{y}$ the vector $\vec{B}$ can be written as, $- 3.0 \overset{⏜}{i} - 3.0 \overset{⏜}{j} - 4.0 \overset{⏜}{k}$

Step by step solution

Given data

$q = 2 \vec{v} = 2.0 \overset{⏜}{i} + 4.0 \overset{⏜}{j} + 6.0 \overset{⏜}{k} \vec{F} = 4.0 \overset{⏜}{i} - 20 \overset{⏜}{j} + 12 \overset{⏜}{k}$

Understanding the concept

Using the formula for cross-product we can find the vector $\vec{B}$ from the given vectors $\vec{F}$ and $\vec{v}$ .

The vector product is written as,

$(\vec{a} \times \overset{⇀}{b}) = (a_{y} b_{z} - b_{y} a_{z}) \overset{⏜}{i} + (a_{z} b_{x} - b_{z} a_{x}) \overset{⏜}{j} + (a_{x} b_{y} - b_{y} a_{x}) \overset{⏜}{k}$ (i)

Calculate B→ in unit-vector notation

Use equation (i) to write the cross product between $\vec{v}$ and $\vec{B}$ .

$\vec{F} = q \vec{v} \times \vec{B} F_{x} \overset{⏜}{i} + F_{y} \overset{⏜}{j} + F_{z} \overset{⏜}{k} = q (v_{y} B_{z} - v_{z} B_{y}) \overset{⏜}{i} + q (v_{z} B_{x} - v_{x} B_{z}) \overset{⏜}{j} + q (v_{x} B_{y} - v_{y} B_{x}) \overset{⏜}{k}$

Comparing the above equation with the given vector $\vec{F}$ , we get

$2 (4 B_{z} - 6 B_{y}) = 4 2 (6 B_{x} - 2 B_{z}) = - 2 2 (2 B_{y} - 4 B_{x}) = 12$

As $B_{x} = B_{y}$ , above equation becomes,

$2 (2 B_{y} - 4 B_{y}) = 12 4 B_{y} - 8 B_{y} = 12 - 4 B_{y} = 12 B_{y} = - 3$

Inserting this value in above equation we get,

$2 (4 B_{z} - 6 (- 3)) = 4 B_{z} = - 4$

Calculate the value of $B_{x}$ from the above equations.

$B_{x} = B_{y} B_{x} = - 3$

Therefore, the vector $\vec{B}$ can be written as $- 3.0 \overset{⏜}{i} - 3.0 \overset{⏜}{j} - 4.0 \overset{⏜}{k}$

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In the product $\vec{F} = q \vec{v} \times \vec{B}$ , take $q = 2, \vec{v} = 2.0 \overset{⏜}{i} + 4.0 \overset{⏜}{j} + 60 \overset{⏜}{k}$ ,and $\vec{F} = 4.0 \overset{⏜}{i} - 20 \overset{⏜}{j} + 12 \overset{⏜}{k}$ . What then is $\vec{B}$ in unit-vector notation if $B_{x} = B_{y}$ ?

Short Answer

Step by step solution

Given data

Understanding the concept

Calculate B→ in unit-vector notation

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In the product F→=qv→×B→, takeq=2,v→=2.0i⏜+4.0j⏜+60k⏜,andF→=4.0i⏜-20j⏜+12k⏜. What then isB→in unit-vector notation ifBx=By?

Short Answer

Step by step solution

Given data

Understanding the concept

Calculate B→ in unit-vector notation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Force

Medical Physics

Particle Model of Matter

Nuclear Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

In the product $\vec{F} = q \vec{v} \times \vec{B}$ , take $q = 2, \vec{v} = 2.0 \overset{⏜}{i} + 4.0 \overset{⏜}{j} + 60 \overset{⏜}{k}$ ,and $\vec{F} = 4.0 \overset{⏜}{i} - 20 \overset{⏜}{j} + 12 \overset{⏜}{k}$ . What then is $\vec{B}$ in unit-vector notation if $B_{x} = B_{y}$ ?