A rectangular loop is situated in a region with a uniform magnetic field of \(0.1 \mathrm{~T}\) pointing into the page, as shown below. The length of the loop is \(20 \mathrm{~cm}\) and the width is \(10 \mathrm{~cm}\). a) What is the magnetic flux through the loop? b) If the value of \(B\) is increased from \(0.1 \mathrm{~T}\) to \(0.5 \mathrm{~T}\) in \(0.3 \mathrm{~s}\), what will be the EMF induced into the loop? What will be the direction of the induced current? Justify your answer. c) If the resistance of the loop is \(R=1 \Omega\), what is the value of the current? d) If the \(B\)-field is tilted at an angle \(\theta=15^{\circ}\) from the normal to the page, what is the flux through the loop? e) Give two ways that one could change the induced EMF in the loop, other than changing \(B\). f) Draw a rotation axis from left to right in the plane of the page that passes through the loop's center. The loop is rotated about this axis at a constant angular velocity. Write an expression for the flux in terms of the angular velocity \(\omega\) and the time \(t\).

Step by step solution

01

Calculate the magnetic flux

Use the formula for magnetic flux: Φ = B⋅A⋅cosθ Here, B = 0.1 T, A = (20 cm × 10 cm) = 0.2 m × 0.1 m = 0.02 m², and θ = 0° because the magnetic field is perpendicular to the loop. So, cosθ = cos(0°) = 1 Φ = (0.1 T) × (0.02 m²) × 1 = 0.002 Wb The magnetic flux through the loop is 0.002 Wb. #b) If the value of B is increased from 0.1 T to 0.5 T in 0.3 s, what will be the EMF induced into the loop? What will be the direction of the induced current? Justify your answer.#

02

Calculate the induced EMF

Use Faraday's law of electromagnetic induction: EMF = -ΔΦ/Δt First, find the new magnetic flux. Since θ remains the same throughout, Φ2 = B2⋅A⋅cosθ B2 = 0.5 T, Φ2 = (0.5 T) × (0.02 m²) × 1 = 0.01 Wb. ΔΦ = Φ2 - Φ1 = 0.01 - 0.002 = 0.008 Wb Δt = 0.3 s EMF = -(0.008 Wb) / (0.3 s) = -0.0267 V The induced EMF is approximately -0.0267 V.

03

Find the direction of the induced current

As per Lenz's Law, the direction of the induced current will be such that it opposes the change in magnetic flux. Since the magnetic field is increasing (0.1 T to 0.5 T) and pointing into the page, the induced current will create a magnetic field pointing out of the page to oppose the change. This will result in a counter-clockwise current direction. #c) If the resistance of the loop is R=1 Ω, what is the value of the current?#

04

Calculate the current

Use Ohm's Law: I = V/R I = (-0.0267 V) / (1 Ω) = -0.0267 A The value of the current is approximately -0.0267 A. #d) If the B-field is tilted at an angle θ=15° from the normal to the page, what is the flux through the loop?#

05

Calculate the flux with the tilted magnetic field

First, we need to find the component of the magnetic field that is perpendicular to the loop (since that is the one contributing to the flux). B⊥ = B⋅cosθ = 0.5⋅cos(15°) Φ = B⊥⋅A = (0.5⋅cos(15°)) × (0.02 m²) = 0.00967 Wb The flux through the loop with a tilted magnetic field is approximately 0.00967 Wb. #e) Give two ways that one could change the induced EMF in the loop, other than changing B.# 1. Change the area of the loop: By increasing or decreasing the size of the loop, the magnetic flux through it will change, thus changing the induced EMF. 2. Change the angle between the magnetic field and the loop: By tilting the loop, we can directly change the angle θ, impacting the magnetic flux and the induced EMF. #f) Write an expression for the flux in terms of the angular velocity ω and the time t.

06

Write the expression for flux in terms of angular velocity and time

As the loop rotates through angle θ(t) = ωt due to constant angular velocity ω, where θ(t) will represent the angle between the magnetic field and the loop normal at time t. Φ(t) = B⋅A⋅cos(ωt) The flux as a function of time and angular velocity is given by Φ(t) = B⋅A⋅cos(ωt).

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