Question

A wire of length \(\ell=10.0 \mathrm{~cm}\) is moving with constant velocity in the \(x y\) -plane; the wire is parallel to the \(y\) -axis and moving along the \(x\) -axis. If a magnetic field of magnitude \(1.00 \mathrm{~T}\) is pointing along the positive \(z\) -axis, what must the velocity of the wire be in order for a potential difference of \(2.00 \mathrm{~V}\) to be induced across it?

Short answer

Answer: The required velocity of the wire is 20 m/s.

Step-by-step solution

Identify the formula used to calculate induced emf in a moving conductor.

In the presence of a uniform magnetic field, the induced emf in a straight conductor of length (l) moving with velocity (v) perpendicular to the field is given by: $$\text{emf} = B l v$$ Where B is the magnetic field and v is the velocity of the conductor.

Note down the given values.

We are given the following values: - Wire length, $$l = 10.0\,\text{cm}$$ - Magnetic field, $$B = 1.00\,\text{T}$$ - Induced emf (potential difference), $$\text{emf} = 2.00\,\text{V}$$

Convert length from cm to m.

As SI unit of length is meter(m), we must convert the wire's length from centimeters to meters. $$l = 10.0\,\text{cm} \times \frac{1\,\text{m}}{100\,\text{cm}} = 0.1\,\text{m}$$

Rearrange the formula and find the velocity.

We have to find the velocity (v) of the wire. We can rearrange the formula to solve for v: $$v = \frac{\text{emf}}{B l}$$ Now, plug in the given values: $$v = \frac{2.00\,\text{V}}{(1.00\,\text{T})(0.1\,\text{m})} = 20\,\text{m/s}$$

State the final answer.

The required velocity of the wire in order for a potential difference of 2.00 V to be induced across it is: $$v = 20\,\text{m/s}$$

Key concepts

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A common source of a magnetic field is a magnet. Magnetic fields are represented by magnetic field lines, with the field direction pointing from the north to the south pole of a magnet, and the density of the field lines corresponding to the strength of the magnetic field.

In our exercise, the magnetic field of magnitude 1.00 T (tesla) is pointing along the positive z-axis. The vector nature of the magnetic field is crucial because it determines the direction of the induced electromotive force (emf) and current according to the right-hand rule: if you point the thumb of your right hand in the direction of the wire's velocity and the fingers in the direction of the magnetic field, the emf is directed out of the palm.

Potential Difference

Potential difference, also known as voltage, between two points is the work done to move a unit charge from one point to the other. It is measured in volts (V). In the context of electromagnetism, a potential difference can be induced across a conductor when it is moving through a magnetic field or when the magnetic field through a loop changes.

In the given problem, a potential difference of 2.00 V is induced across the wire due to its motion in the magnetic field. This induced potential difference is a manifestation of an induced emf, which is caused by a change in the magnetic environment of the wire. The potential difference drives a current if there is a closed loop for it to flow through.

Velocity of a Conductor

The velocity of a conductor refers to its speed in a given direction. In the scenario of electromagnetic induction, the velocity is critical when the conductor is moving relative to a magnetic field. The magnitude and direction of the velocity influence the magnitude of the induced emf. The greater the velocity at which the conductor moves through the magnetic field, the greater the potential difference induced across it.

The wire in the exercise is moving with a constant velocity along the x-axis. To find out what this velocity needs to be for an induced potential difference of 2.00 V, we use Faraday's law of induction. The resulting calculation shows that the required velocity is 20 m/s, meaning the wire must move at this speed to generate the required potential difference.

Faraday's Law of Induction

Faraday's law of induction is a fundamental principle in electromagnetism, predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induction. Faraday's law states that the induced emf in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

In the simplest form, when a straight conductor moves through a uniform magnetic field, the induced emf (\text{emf}) can be calculated by the formula: \[\text{emf} = B l v\] where \(B\) is the magnetic field strength, \(l\) is the length of the conductor, and \(v\) is the velocity of the conductor. This relationship allows us to solve for unknown variables, as seen in this exercise where the velocity needed for a specific induced potential difference is calculated.