Q.If the magnetic field is parallel to the positive $y$-axis and the charged particle is moving along the positive $x$-axis (Fig. 4.4), which way would the Lorentz force be for

The key idea is the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$. The direction of the force depends on the sign of the charge $q$.

Step 1: The magnetic field $\vec{B}$ is along $+y$, and the velocity $\vec{v}$ is along $+x$. Using the right-hand rule for $\vec{v} \times \vec{B}$: point fingers along $+x$, curl them toward $+y$. The thumb points along $+z$ (upward). This is the direction of the force for a positive charge.

Step 2: For a proton ($q > 0$), the force is directly along $+z$ (upward). …

The core idea here is the direction of the Lorentz force, set by the cross product $\vec{v}\times\vec{B}$ together with the sign of the charge. The Lorentz force law tells us that a charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the field. This perpendicular nature is what makes magnetic forces so elegant (and tricky): they never speed up or slow down a particle, only bend its path.

For this problem, we have:

• $\vec{B}$ along $+y$ (positive y-axis)
• $\vec{v}$ along $+x$ (positive x-axis)
• Two cases: $q = +e$ (proton) and $q = -e$ (electron)

The right-hand rule (or the cross product) gives the direction of $\vec{v} \times \vec{B}$, and then the sign of $q$ decides whether the force is along that direction or opposite to it.

1. Find the direction of $\vec{v} \times \vec{B}$

   Using the right-hand rule: point fingers along $\vec{v}$ (positive $x$), curl them toward $\vec{B}$ (positive $y$). Your thumb points along $+z$ (upward).

   Mathematically: $\hat{x} \times \hat{y} = \hat{z}$, so $\vec{v} \times \vec{B}$ points along $+z$.

2. Apply the Lorentz force law

   The force is $\vec{F} = q(\vec{v} \times \vec{B})$.

   • For a proton ($q = +e$): $\vec{F} = +e(\text{along }+z)$, so the force is along $+z$.
   • For an electron ($q = -e$): $\vec{F} = -e(\text{along }+z)$, so the force is along $-z$. …

Method: Right-Hand Rule (for Lorentz Force)

This is a vector cross-product problem:

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

Steps

1. Identify the directions

   • $\vec{B}$ is along +y
   • $\vec{v}$ is along +x
   • For a positive charge, use the right hand directly
   • For a negative charge, reverse the direction

2. Apply the right-hand rule for positive charge

   • Point fingers of right hand along $\vec{v}$ (+x)
   • Curl fingers toward $\vec{B}$ (+y)
   • Thumb points along $\vec{v} \times \vec{B}$ → +z direction (out of page)

3. Adjust for sign of charge

   • Proton ($q > 0$): force is along +z (out of page)
   • Electron ($q < 0$): force is opposite → −z (into page)

Final Answer

| Particle | Charge | Force direction | …

Here are the common mistakes students make when solving this Magnetic Force Balance problem, along with how to avoid each.

Mistake 1: Using the wrong hand rule for the charge sign

The error:

Students often use the right-hand rule for both positive and negative charges without flipping the direction for negative charges.

How to avoid:

• For a positive charge (proton): Use the right-hand rule. Point fingers along velocity ($+x$), curl them toward magnetic field ($+y$), thumb gives force direction ($+z$).
• For a negative charge (electron): Use the right-hand rule to find the direction for a positive charge, then reverse it.
• Alternatively, use the left-hand rule directly for negative charges.

Key result:

• Proton: force along $+z$ (out of page).
• Electron: force along $-z$ (into page).

Mistake 2: Forgetting the cross product order

The error:

Writing $\vec{F} = q(\vec{B} \times \vec{v})$ instead of the correct $\vec{F} = q(\vec{v} \times \vec{B})$.

How to avoid:

• Memorise: Lorentz force is $\vec{F} = q(\vec{v} \times \vec{B})$.
• The cross product $\vec{v} \times \vec{B}$ is not the same as $\vec{B} \times \vec{v}$ — they give opposite directions.
• Use the right-hand rule with velocity first, then magnetic field.

Mistake 3: Misidentifying the coordinate axes

The error:

Confusing which axis is $x$, $y$, or $z$, especially in 3D diagrams.

How to avoid:

• Draw a clear 3D coordinate system before starting.
• Standard convention:
  • $+x$: right
  • $+y$: up
  • $+z$: out of the page (toward you)
• In this problem:
  • Velocity: $+x$ (right)
  • Magnetic field: $+y$ (up)
  • Force: $+z$ (out of page) for positive charge.

Mistake 4: Ignoring the charge sign when applying the right-hand rule

The error:

Applying the right-hand rule and forgetting to reverse the thumb direction for a negative charge.

How to avoid:

• Always first find the force direction for a positive charge using the right-hand rule.
• Then, if the charge is negative, simply reverse that direction.
• This two-step method prevents sign errors.

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