Question

A conducting rod \(P Q\) of length \(4 \ell\) is rotated about a point \(O\) in a uniform magnetic field \(\mathrm{B} \rightarrow \mathrm{PO}=\ell\) Then (a) \(\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{P}}=-\left[\left\\{\mathrm{B} \omega \ell^{2}\right\\} /\right.\) \(\\{2\\}] \quad\) (b) \(\mathrm{V}_{\mathrm{Q}}-\mathrm{V}_{\mathrm{O}}=(5 / 2) \mathrm{B} \operatorname{co} \ell^{2}\) (c) \(\mathrm{V}_{Q}-\mathrm{V}_{\mathrm{O}}=(9 / 2) \mathrm{B} \omega \ell^{2}\) (d) \(\mathrm{V}_{\mathrm{P}}-\mathrm{V}_{\mathrm{Q}}=4 \mathrm{~B} \omega \ell^{2}\)

Short answer

The short answer is: (c) V_Q - V_O = (9 / 2) Bωℓ^2.

Step-by-step solution

Find the tangential velocities

To find the tangential velocities at points P and Q, we'll use the formula: v = ωr where ω is the angular velocity of the rod about point O, and r is the distance from point O to the point of interest (P or Q). For point P: u = ω(ℓ) For point Q: v = ω(5ℓ)

Calculate the emf induced in the rod

Using the formula given for the emf induced in a rotating rod in a magnetic field: emf = Bℓ(v - u) We substitute the tangential velocities found in step 1: emf = Bℓ(ω(5ℓ) - ω(ℓ)) emf = Bℓω(5ℓ - ℓ) emf = 4Bℓ^2ω

Determine the potential difference between points P and Q, O and Q

The potential difference between points P and Q is the emf: V_Q - V_P = 4Bℓ^2ω Now, let's find the potential difference between points O and Q: V_Q - V_O = V_Q - (V_P + 4Bℓ^2ω) Using the relation found above: V_Q - V_P = 4Bℓ^2ω, V_Q - V_O = 9Bℓ^2ω / 2

Compare the results with given options

Comparing the results from steps 3 with the given options, we can see that: (a) is incorrect because the sign and fraction are not correct. (b) is incorrect because co is not part of the expression. (c) is correct as it matches the expression for the potential difference between points Q and O derived in our solution. (d) is incorrect because the expression is for potential difference between points P and Q and the factor is not correct. So, the correct answer is (c) V_Q - V_O = (9 / 2) Bωℓ^2.

Key concepts

Rotational motion

Rotational motion involves objects that are rotating, meaning they spin around an axis. Think of a rod being rotated around a fixed point, much like a merry-go-round spins around its center. In the exercise, the rod is rotating about point O. This kind of motion is characterized by angular velocity, denoted as \( \omega \), which tells us how fast the object is spinning.For rotational motion, each point on the object moves in a circular path. The speed of a point depends on its distance from the axis. This speed is called tangential velocity, and it can be found using the formula:- \( v = \omega r \), where: - \( \omega \) is the angular velocity, - \( r \) is the distance from the point to the axis.

In our problem, for example, point P's tangential velocity is \( \omega \ell \), and point Q's is \( \omega(5\ell) \). As the rod spins, these velocities help us understand how quickly different parts of the rod are moving due to the rotation.

Magnetic fields

Magnetic fields are invisible forces that exert influence on moving charges and conductive materials. Picture it as a force field around a magnet where it can attract certain metals. In the exercise, a uniform magnetic field affects the rotating rod. The interaction between the magnetic field and the moving rod leads to electromagnetic induction, a process where a voltage (or emf) is generated. This happens because the magnetic field causes electrons in the rod to experience an electric force, leading them to move. Here are some key points about magnetic fields:- **Uniform magnetic fields**: All lines of force are parallel and evenly spaced.- **Electromagnetic induction**: Generates electrical current through the movement of a conductor in a magnetic field.When the rod rotates, it cuts through the magnetic lines, inducing an emf according to the formula:- \( \text{emf} = B \ell (v - u) \). This formula tells us how much voltage is induced based on the length of the rod (\( \ell \)), the difference in tangential velocities (\( v - u \)), and the strength of the magnetic field (\( B \)).

Potential difference

Potential difference, often known as voltage, refers to the amount of energy required to move a unit charge between two points in an electric field. Think of it like the pressure needed to push water through a pipe.In our exercise, we calculate the potential difference between different points on the rod (P, Q, and O). Using the induced emf from the rotation, we can find the potential difference by employing the emf as a direct measure.- **The potential difference between P and Q**: This is understood directly from the total emf generated by the rod. Based on our calculation earlier: - \( V_Q - V_P = 4B\ell^2\omega \)- **The potential difference between O and Q**: This includes additional considerations due to the rotational dynamics: - \( V_Q - V_O = \frac{9}{2} B \ell^2 \omega \)Voltage is a fundamental concept in understanding how electric forces cause movement in circuits or across conductive materials. It's essential for powering devices and is a cornerstone of electrical engineering.