A conductor carries a current vertically downwards in a magnetic field directed southwards, It then experiences a force in the direction of:

Question

Prepp Team · Accepted Answer

Understanding the Force on a Current-Carrying Conductor

When a conductor carrying electric current is placed in a magnetic field, it experiences a force. The direction of this force depends on the direction of the current and the direction of the magnetic field. This phenomenon is governed by the principles of electromagnetism.

To determine the direction of the force, we can use Fleming's Left-Hand Rule or the vector cross product formula for the Lorentz force.

Applying Fleming's Left-Hand Rule

Fleming's Left-Hand Rule is a common mnemonic used to find the direction of the force, magnetic field, or current when the directions of the other two quantities are known. For a current-carrying conductor in a magnetic field:

The Forefinger points in the direction of the external magnetic field (\(\vec{B}\)).
The Middle finger points in the direction of the conventional current (\(I\)).
The Thumb then points in the direction of the force (\(\vec{F}\)) experienced by the conductor.

These three fingers must be held mutually perpendicular to each other.

Step-by-Step Application of the Rule

Let's apply Fleming's Left-Hand Rule to the specific conditions given in the problem:

The current is vertically downwards. So, point your left hand's middle finger straight down towards the ground.
The magnetic field is directed southwards. While keeping your middle finger pointing down, align your left hand's forefinger to point towards the South direction.
Now, observe the direction in which your left hand's thumb is pointing. With the middle finger pointing down and the forefinger pointing South, your thumb will be pointing towards the West direction.

Therefore, the force on the conductor is directed towards the West.

Alternative Method: Lorentz Force Formula

The magnetic force (\(\vec{F}\)) on a current-carrying conductor segment of length \(\vec{L}\) in a magnetic field \(\vec{B}\) is given by the formula:

\(\vec{F} = I(\vec{L} \times \vec{B})\)

Here, \(I\) is the current, \(\vec{L}\) is a vector in the direction of the current with a magnitude equal to the length of the conductor, and \(\vec{B}\) is the magnetic field vector. The direction of the force is given by the direction of the cross product \(\vec{L} \times \vec{B}\).

Let's define a coordinate system:

Let North be along the +y axis. South is along the -y axis.
Let East be along the +x axis. West is along the -x axis.
Let Up be along the +z axis. Down be along the -z axis.

Based on the problem description:

Current direction is downwards, so the vector \(\vec{L}\) is in the \(-\hat{k}\) direction.
Magnetic field direction is southwards, so the vector \(\vec{B}\) is in the \(-\hat{j}\) direction.

Now, let's calculate the direction of the force using the cross product \(\vec{L} \times \vec{B}\):

\(\vec{F} \propto \vec{L} \times \vec{B} \propto (-\hat{k}) \times (-\hat{j})\)

\(\vec{F} \propto (\hat{k}) \times (\hat{j})\)

Using the standard rules for cross products of unit vectors (\(\hat{i} \times \hat{j} = \hat{k}\), \(\hat{j} \times \hat{k} = \hat{i}\), \(\hat{k} \times \hat{i} = \hat{j}\)), we know that \(\hat{k} \times \hat{j} = -\hat{i}\).

Thus, the force \(\vec{F}\) is proportional to \(-\hat{i}\). In our coordinate system, the \(-\hat{i}\) direction corresponds to the West direction.

Conclusion on Force Direction

Both Fleming's Left-Hand Rule and the Lorentz force calculation lead to the same result: the force on the conductor carrying current vertically downwards in a magnetic field directed southwards is towards the West.

Quantity	Direction Given	Rule/Vector Result	Direction
Current (\(I\))	Vertically Downwards	Middle Finger / \(-\hat{k}\)	Down
Magnetic Field (\(\vec{B}\))	Southwards	Forefinger / \(-\hat{j}\)	South
Force (\(\vec{F}\))	Resulting Direction	Thumb / \(-\hat{i}\)	West

Revision Table: Electromagnetism Concepts

Concept	Description	Relation
Current (\(I\))	Flow of electric charge.	Conventional direction is flow of positive charge.
Magnetic Field (\(\vec{B}\))	Region where magnetic forces are exerted.	Has both magnitude and direction (Vector).
Magnetic Force (\(\vec{F}\))	Force on charge or current in a magnetic field.	Perpendicular to both velocity/current and magnetic field.
Fleming's Left-Hand Rule	Mnemonic to find the direction of force, field, or current in motors.	Thumb: Force, Forefinger: Field, Middle: Current.
Lorentz Force Formula	Mathematical description of magnetic force.	\(\vec{F} = q(\vec{v} \times \vec{B})\) or \(\vec{F} = I(\vec{L} \times \vec{B})\).

Additional Information: Related Concepts

The interaction between moving charges (or currents) and magnetic fields is a fundamental principle in physics. This interaction gives rise to the magnetic force. The direction of this force is always perpendicular to both the velocity of the charge (or the direction of the current) and the direction of the magnetic field. This is why the cross product is used in the formula.

The magnitude of the force depends on the strength of the current, the length of the conductor (or the velocity of the charge), the strength of the magnetic field, and the angle between the current/velocity vector and the magnetic field vector (\(|\vec{F}| = I|\vec{L}||\vec{B}|\sin\theta\)). The force is maximum when the current is perpendicular to the magnetic field (\(\theta = 90^\circ\)).
Fleming's Right-Hand Rule is used for generators and determines the direction of induced current, while the Left-Hand Rule is for motors and determines the direction of force.

Understanding these directional rules is essential for studying motors, generators, and other electromagnetic devices.