Question

A metal blade of length L = 300 cm spins at a constant rate of 17 rad/s about an axis that is perpendicular to the blade and through its center. A uniform magnetic field B = 4.0 mT is perpendicular to the plane of rotation. What is the magnitude of the potential difference (in V) between the center of the blade and either of its ends?

Answer 1

We are being given that:

• The length of a metal blade = 300 cm
• The angular velocity at which the metal blade is rotating about its axis  is ω = 17 rad/s
• The magnetic field B = 4.0 mT

A pictorial view showing the diagrammatic representation of the information given in the question is being attached in the image below.

From the attached image below, the potential difference across the conducting element of the length (dx) moving with the velocity (v) appears to be perpendicular to the magnetic field (B).

The magnitude of the potential difference induced between the center of the blade in relation to either of its ends can be determined by using the derived formula from Faraday's law of induction which can be expressed as:

$\mathsf{E = B\times l\times v}$

where;

B = magnetic field

l = length

v = relative speed

From the diagram, Let consider the length of the conducting element (dx) at a distance of length (x) from the center O.

Then, the velocity (v) = ωx

The potential difference across it can now be expressed as:

$\mathsf{dE = B*(dx)*(\omega x)}$

For us to determine the potential difference, we need to carry out the integral form from center point O to L/2.

∴

$\mathsf{E = \int ^{L/2}_{0}* B (\omega x ) *(dx)}$

$\mathsf{E = B (\omega ) \times \Big[ \dfrac{x^2}{2}\Big]^{L/2}_{0}}$

$\mathsf{E = B (\omega ) * \Big[ \dfrac{L^2}{8}\Big]}$

Recall that,

magnetic field (B) = 4 mT = 4 × 10⁻³  T

Length L = 300 cm = 3m

angular velocity (ω) = 17 rad/s

$\mathsf{E = (4\times 10^{-3}) * (17) \Big[ \dfrac{(1.5)^2}{8}\Big]}$

$\mathsf{E = 19.13 mV}$

Thus, we can now conclude that the magnitude of the potential difference as a result of the rotation caused by the metal blade from the center to either of its ends is 19.13 mV.

Learn more about Faraday's law of induction here:

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