Question:
A positive point charge with velocity \(\vec v=5\hat x\) enters a region having electric field \(\vec E=4\hat y\) and magnetic field \(\vec B=-6\hat z\). Which one of the following statements is correct for the force on the charge as it enters the region?
A positive point charge with velocity \(\vec v=5\hat x\) enters a region having electric field \(\vec E=4\hat y\) and magnetic field \(\vec B=-6\hat z\). Which one of the following statements is correct for the force on the charge as it enters the region?
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Apply F = q(E + v cross B), compute v cross B first, then compare the resulting direction with E and with B.
Updated On: Jul 20, 2026
- The force will be along the magnetic field but perpendicular to the electric field
- The force will be along the electric field but perpendicular to the magnetic field
- The force will be perpendicular to both electric and magnetic field
- The magnetic field does not exert any force on the charge
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The Correct Option is B
Solution and Explanation
Step 1: Recall the Lorentz force law.
The total electromagnetic force on a charge \(q\) moving with velocity \(\vec v\) in fields \(\vec E\) and \(\vec B\) is
\[ \vec F=q\left(\vec E+\vec v\times\vec B\right) \]
Step 2: Compute \(\vec v\times\vec B\) using the determinant method.
\[ \vec v\times\vec B=\begin{vmatrix}\hat x&\hat y&\hat z\\5&0&0\\0&0&-6\end{vmatrix} \]
Expanding: the \(\hat x\) component is \((0)(-6)-(0)(0)=0\); the \(\hat y\) component is \(-[(5)(-6)-(0)(0)]=30\); the \(\hat z\) component is \((5)(0)-(0)(0)=0\). So
\[ \vec v\times\vec B=30\,\hat y \]
Step 3: Add the electric force term.
\[ \vec F=q\left(4\hat y+30\hat y\right)=q(34\hat y)=34q\,\hat y \]
Step 4: Interpret the direction of the total force.
The net force \(34q\hat y\) points purely along \(\hat y\), which is exactly the direction of the electric field \(\vec E=4\hat y\), so the force is along the electric field. Since \(\hat y\) is perpendicular to \(\hat z\), the direction of \(\vec B\), the force is also perpendicular to the magnetic field.
Step 5: Rule out the other options.
Option (A) says the force is along \(\vec B\) (the \(\hat z\) direction), but the computed force has zero \(\hat z\) component, so (A) is wrong. Option (C) says the force is perpendicular to both fields, but the force is actually parallel to \(\vec E\), not perpendicular to it, so (C) is wrong. Option (D) says the magnetic field exerts no force, but the term \(\vec v\times\vec B=30\hat y\) is clearly nonzero; it only happens that this magnetic force adds constructively along the same direction as the electric force, which does not mean the magnetic force is absent, so (D) is wrong.
Step 6: Final Answer.
\[ \boxed{\text{Force is along}\ \vec E,\ \text{perpendicular to}\ \vec B} \]
The total electromagnetic force on a charge \(q\) moving with velocity \(\vec v\) in fields \(\vec E\) and \(\vec B\) is
\[ \vec F=q\left(\vec E+\vec v\times\vec B\right) \]
Step 2: Compute \(\vec v\times\vec B\) using the determinant method.
\[ \vec v\times\vec B=\begin{vmatrix}\hat x&\hat y&\hat z\\5&0&0\\0&0&-6\end{vmatrix} \]
Expanding: the \(\hat x\) component is \((0)(-6)-(0)(0)=0\); the \(\hat y\) component is \(-[(5)(-6)-(0)(0)]=30\); the \(\hat z\) component is \((5)(0)-(0)(0)=0\). So
\[ \vec v\times\vec B=30\,\hat y \]
Step 3: Add the electric force term.
\[ \vec F=q\left(4\hat y+30\hat y\right)=q(34\hat y)=34q\,\hat y \]
Step 4: Interpret the direction of the total force.
The net force \(34q\hat y\) points purely along \(\hat y\), which is exactly the direction of the electric field \(\vec E=4\hat y\), so the force is along the electric field. Since \(\hat y\) is perpendicular to \(\hat z\), the direction of \(\vec B\), the force is also perpendicular to the magnetic field.
Step 5: Rule out the other options.
Option (A) says the force is along \(\vec B\) (the \(\hat z\) direction), but the computed force has zero \(\hat z\) component, so (A) is wrong. Option (C) says the force is perpendicular to both fields, but the force is actually parallel to \(\vec E\), not perpendicular to it, so (C) is wrong. Option (D) says the magnetic field exerts no force, but the term \(\vec v\times\vec B=30\hat y\) is clearly nonzero; it only happens that this magnetic force adds constructively along the same direction as the electric force, which does not mean the magnetic force is absent, so (D) is wrong.
Step 6: Final Answer.
\[ \boxed{\text{Force is along}\ \vec E,\ \text{perpendicular to}\ \vec B} \]
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