Solved Objectives: Gain practice in using the Right-Hand | Chegg.com - Free Printable

We are given a series of scenarios involving the magnetic force on a moving charge, governed by the formula:

$$
\vec{F} = q(\vec{v} \times \vec{B})
$$

Where:
- $ \vec{F} $ is the magnetic force,
- $ q $ is the charge (positive or negative),
- $ \vec{v} $ is the velocity of the charge,
- $ \vec{B} $ is the magnetic field,
- $ \times $ denotes the cross product.

The direction of the force depends on:
- The right-hand rule for positive charges,
- The left-hand rule (or reverse of right-hand rule) for negative charges.

We use:
- $ \otimes $: Magnetic field into the page
- $ \odot $: Magnetic field out of the page

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Part a) Find the direction of the force $ \vec{F} $

We are given $ \vec{v} $, $ \vec{B} $, and sign of $ q $. Use $ \vec{F} = q(\vec{v} \times \vec{B}) $

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#### 1. $ q $: Positive, $ \vec{B} $: Into page ($ \otimes $), $ \vec{v} $: Right (→)

- $ \vec{v} $ → right
- $ \vec{B} $ → into page (⊗)
- Use right-hand rule: Point fingers in direction of $ \vec{v} $ (right), curl toward $ \vec{B} $ (into page). Thumb gives $ \vec{v} \times \vec{B} $: downward.
- Since $ q > 0 $, $ \vec{F} $ is downward.

✔ Answer: Down

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#### 2. $ q $: Negative, $ \vec{B} $: Down, $ \vec{v} $: Out of page ($ \odot $)

- $ \vec{v} $: out of page (⊙)
- $ \vec{B} $: down (↓)
- $ \vec{v} \times \vec{B} $: Use right-hand rule:
- Fingers: out of page (⊙)
- Curl toward down (↓) → thumb points to the left (←)
- So $ \vec{v} \times \vec{B} $ → left
- But $ q < 0 $ ⇒ reverse direction → right

✔ Answer: Right

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#### 3. $ q $: Positive, $ \vec{B} $: Left, $ \vec{v} $: Up

- $ \vec{v} $: up (↑)
- $ \vec{B} $: left (←)
- $ \vec{v} \times \vec{B} $: Right-hand rule:
- Fingers: up
- Curl to left → thumb points into the page (⊗)
- $ q > 0 $ ⇒ $ \vec{F} $: into the page

✔ Answer: Into the page (⊗)

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#### 4. $ q $: Negative, $ \vec{B} $: Into page (⊗), $ \vec{v} $: Up

- $ \vec{v} $: up (↑)
- $ \vec{B} $: into page (⊗)
- $ \vec{v} \times \vec{B} $: Right-hand rule:
- Fingers: up
- Curl into page → thumb points to the right (→)
- But $ q < 0 $ ⇒ reverse → left

✔ Answer: Left

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#### 5. $ q $: Positive, $ \vec{B} $: Left, $ \vec{v} $: Into page (⊗)

- $ \vec{v} $: into page (⊗)
- $ \vec{B} $: left (←)
- $ \vec{v} \times \vec{B} $: Right-hand rule:
- Fingers: into page (⊗)
- Curl to left → thumb points up (↑)
- $ q > 0 $ ⇒ $ \vec{F} $: up

✔ Answer: Up

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✔ Part a Summary:

| Case | Charge | B | v | Force |
|------|--------|----|----|-------|
| 1 | + | ⊗ (into) | → | ↓ |
| 2 | – | ↓ | ⊙ (out) | → |
| 3 | + | ← | ↑ | ⊗ (into) |
| 4 | – | ⊗ (into) | ↑ | ← |
| 5 | + | ← | ⊗ (into) | ↑ |

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Part b) Find the direction of $ \vec{B} $, given $ \vec{v}, \vec{F}, q $

Use: $ \vec{F} = q(\vec{v} \times \vec{B}) $ ⇒ $ \vec{B} $ must satisfy this.

We can rearrange: $ \vec{v} \times \vec{B} = \frac{\vec{F}}{q} $, so direction of $ \vec{B} $ is determined via right-hand rule, adjusted for sign.

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#### 1. $ q $: Negative, $ \vec{v} $: Down, $ \vec{F} $: Right

- $ \vec{v} $: down (↓)
- $ \vec{F} $: right (→)
- $ q < 0 $ ⇒ $ \vec{v} \times \vec{B} = \frac{\vec{F}}{q} $ is opposite to $ \vec{F} $, so $ \vec{v} \times \vec{B} $: left (←)
- We have $ \vec{v} $: down, $ \vec{v} \times \vec{B} $: left
- Use right-hand rule: point fingers down, want cross product to be left → palm or curl must give that
- Try: fingers down, thumb left → then curled fingers point into the page (⊗)
- So $ \vec{B} $: into the page (⊗)

✔ Answer: Into the page (⊗)

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#### 2. $ q $: Negative, $ \vec{v} $: Up, $ \vec{F} $: Into page (⊗)

- $ \vec{v} $: up (↑)
- $ \vec{F} $: into page (⊗)
- $ q < 0 $ ⇒ $ \vec{v} \times \vec{B} = \frac{\vec{F}}{q} $ ⇒ opposite to $ \vec{F} $ ⇒ out of page (⊙)
- So $ \vec{v} \times \vec{B} $: ⊙
- $ \vec{v} $: up, $ \vec{v} \times \vec{B} $: out of page
- Right-hand rule: fingers up, thumb out of page → curled fingers point right
- So $ \vec{B} $: right

✔ Answer: Right

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#### 3. $ q $: Positive, $ \vec{v} $: Out of page (⊙), $ \vec{F} $: Right

- $ \vec{v} $: ⊙
- $ \vec{F} $: →
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} = \vec{F} $: →
- $ \vec{v} $: out of page, $ \vec{v} \times \vec{B} $: right
- Right-hand rule: fingers out of page, thumb right → curled fingers point down
- So $ \vec{B} $: down

✔ Answer: Down

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#### 4. $ q $: Positive, $ \vec{v} $: Into page (⊗), $ \vec{F} $: Down

- $ \vec{v} $: ⊗
- $ \vec{F} $: ↓
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} $: ↓
- $ \vec{v} $: into page, $ \vec{v} \times \vec{B} $: down
- Right-hand rule: fingers into page, thumb down → curled fingers point left
- So $ \vec{B} $: left

✔ Answer: Left

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#### 5. $ q $: Positive, $ \vec{v} $: ?, $ \vec{F} $: Left, $ \vec{B} $: Into page (⊗)

Wait — we're asked to find $ \vec{v} $, not $ \vec{B} $. Wait! This is part c).

Hold on — let's correct.

This is part b), but last one says:

> $ q $: Positive, $ \vec{v} $: ? , $ \vec{F} $: Left, $ \vec{B} $: Into page?

No — actually, in part b), it says:

> "Find the direction of the magnetic field..."

But in the fifth diagram of part b):

- $ q $: Positive
- $ \vec{v} $: ⊙ (out of page)
- $ \vec{F} $: ← (left)

So we need to find $ \vec{B} $

Let’s fix:

#### 5. $ q $: Positive, $ \vec{v} $: Out of page (⊙), $ \vec{F} $: Left (←)

- $ \vec{v} $: ⊙
- $ \vec{F} $: ←
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} = \vec{F} $: ←
- $ \vec{v} $: out of page, $ \vec{v} \times \vec{B} $: left
- Right-hand rule: fingers out of page, thumb left → curled fingers point down
- So $ \vec{B} $: down

✔ Answer: Down

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✔ Part b Summary:

| Case | Charge | v | F | B |
|------|--------|----|----|----|
| 1 | – | ↓ | → | ⊗ (into) |
| 2 | – | ↑ | ⊗ (into) | → |
| 3 | + | ⊙ (out) | → | ↓ |
| 4 | + | ⊗ (into) | ↓ | ← |
| 5 | + | ⊙ (out) | ← | ↓ |

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Part c) Find the direction of $ \vec{v} $, given $ \vec{B}, \vec{F}, q $

Use $ \vec{F} = q(\vec{v} \times \vec{B}) $

Solve for $ \vec{v} $ using right-hand rule.

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#### 1. $ q $: Positive, $ \vec{B} $: Up, $ \vec{F} $: Right

- $ \vec{B} $: ↑
- $ \vec{F} $: →
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} = \vec{F} $: →
- $ \vec{B} $: up, $ \vec{v} \times \vec{B} $: right
- Use right-hand rule: fingers in direction of $ \vec{v} $, curl toward $ \vec{B} $, thumb gives $ \vec{v} \times \vec{B} $
- Want $ \vec{v} \times \vec{B} $: right, $ \vec{B} $: up
- So: point fingers in direction of $ \vec{v} $, curl toward up, thumb → right
- Try: $ \vec{v} $: into the page (⊗)
- Fingers: into page, curl upward → thumb points right ✔
- So $ \vec{v} $: into the page (⊗)

✔ Answer: Into the page (⊗)

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#### 2. $ q $: Negative, $ \vec{B} $: Into page (⊗), $ \vec{F} $: Down

- $ \vec{B} $: ⊗
- $ \vec{F} $: ↓
- $ q < 0 $ ⇒ $ \vec{v} \times \vec{B} = \frac{\vec{F}}{q} $ ⇒ opposite to $ \vec{F} $ ⇒ up (↑)
- So $ \vec{v} \times \vec{B} $: ↑
- $ \vec{B} $: into page (⊗), $ \vec{v} \times \vec{B} $: up
- Right-hand rule: fingers in direction of $ \vec{v} $, curl toward $ \vec{B} $ (into page), thumb up
- Try: $ \vec{v} $: right
- Fingers: right, curl into page → thumb up ✔
- So $ \vec{v} $: right

✔ Answer: Right

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#### 3. $ q $: Positive, $ \vec{B} $: Down, $ \vec{F} $: Left

- $ \vec{B} $: ↓
- $ \vec{F} $: ←
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} $: ←
- $ \vec{B} $: down, $ \vec{v} \times \vec{B} $: left
- Right-hand rule: fingers in direction of $ \vec{v} $, curl toward down, thumb left
- Try: $ \vec{v} $: out of page (⊙)
- Fingers: out of page, curl downward → thumb points left ✔
- So $ \vec{v} $: out of page (⊙)

✔ Answer: Out of the page (⊙)

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#### 4. $ q $: Negative, $ \vec{B} $: Out of page (⊙), $ \vec{F} $: Right

- $ \vec{B} $: ⊙
- $ \vec{F} $: →
- $ q < 0 $ ⇒ $ \vec{v} \times \vec{B} = \frac{\vec{F}}{q} $ ⇒ opposite to $ \vec{F} $ ⇒ left (←)
- So $ \vec{v} \times \vec{B} $: ←
- $ \vec{B} $: out of page, $ \vec{v} \times \vec{B} $: left
- Right-hand rule: fingers in $ \vec{v} $, curl toward out of page, thumb left
- Try: $ \vec{v} $: up
- Fingers: up, curl out of page → thumb left ✔
- So $ \vec{v} $: up

✔ Answer: Up

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#### 5. $ q $: Positive, $ \vec{B} $: Left, $ \vec{F} $: Into page (⊗)

- $ \vec{B} $: ←
- $ \vec{F} $: ⊗
- $ q > 0 $ ⇒ $ \vec{v} \times \vec{B} $: ⊗
- $ \vec{B} $: left, $ \vec{v} \times \vec{B} $: into page
- Right-hand rule: fingers in $ \vec{v} $, curl toward left, thumb into page
- Try: $ \vec{v} $: up
- Fingers: up, curl to left → thumb into page ✔
- So $ \vec{v} $: up

✔ Answer: Up

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✔ Part c Summary:

| Case | Charge | B | F | v |
|------|--------|----|----|----|
| 1 | + | ↑ | → | ⊗ (into) |
| 2 | – | ⊗ (into) | ↓ | → |
| 3 | + | ↓ | ← | ⊙ (out) |
| 4 | – | ⊙ (out) | → | ↑ |
| 5 | + | ← | ⊗ (into) | ↑ |

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✔ Final Answers:

#### a) Direction of force $ \vec{F} $:
1. Down
2. Right
3. Into the page (⊗)
4. Left
5. Up

#### b) Direction of magnetic field $ \vec{B} $:
1. Into the page (⊗)
2. Right
3. Down
4. Left
5. Down

#### c) Direction of velocity $ \vec{v} $:
1. Into the page (⊗)
2. Right
3. Out of the page (⊙)
4. Up
5. Up

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Let me know if you'd like these visualized with diagrams or arrows!