A magnetic field →B=(2πm T/s)t^i exists in a region. In that region, we have a conducting wire of length1 m and resistance 100 Ω. If we can decide the shape to be given to the wire, find the maximum current (in μ A) that can flow in the loop so formed.
Question
A magnetic field →B=(2πm T/s)t^i exists in a region. In that region, we have a conducting wire of length1 m and resistance 100 Ω. If we can decide the shape to be given to the wire, find the maximum current (in μ A) that can flow in the loop so formed.
Solution
The problem involves Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit.
Step 1: Calculate the magnetic flux (Φ) through the loop. The magnetic flux is given by the dot product of the magnetic field (B) and the area vector (A). In this case, the magnetic field is changing with time, so we have B = (2πm T/s)t. The area vector is perpendicular to the direction of the magnetic field and its magnitude is equal to the area of the loop. If we choose the shape of the wire to be a circle, the area of the loop is A = πr^2, where r is the radius of the circle. The length of the wire is given as 1 m, so the circumference of the circle is 2πr = 1 m, which gives r = 1/(2π) m. Substituting this into the equation for A gives A = π/(4π^2) = 1/(4π) m^2. Therefore, the magnetic flux is Φ = B.A = (2πm T/s)t . 1/(4π) = 0.5t m^2T/s.
Step 2: Calculate the induced EMF (ε). According to Faraday's law, the induced EMF is equal to the rate of change of the magnetic flux, i.e., ε = dΦ/dt. Differentiating Φ with respect to t gives ε = d(0.5t)/dt = 0.5 V.
Step 3: Calculate the induced current (I). The induced current is given by Ohm's law, I = ε/R, where R is the resistance of the wire. Substituting the given values gives I = 0.5 V / 100 Ω = 0.005 A = 5 μA.
Therefore, the maximum current that can flow in the loop is 5 μA.
Similar Questions
The loop in the image is moving through the magnetic field, of strength 0.700 T, at a speed of 27.0 m/s. If the resistance in the loop is 1.10 Ω and the width of the wire is x = 9.00 cm, what is the magnitude and direction of the induced current?
The figure below shows two loops of wire having the same axis. The smaller loop has radius a andresistance R and the larger loop has radius b. The smaller loop is above the larger one, by a distance z,which is large compared to the radius b of the larger loop, (z >> b). Hence with current I through the largerloop as indicated, the consequent magnetic field is nearly constant through the plane area bounded bythe smaller loop. Suppose now that z is not constant but is changing at the positive constant rate v z = dz /dt > 0 (z increasing).(a) Determine the magnetic flux across the area bounded by the smaller loop as a function of z.
A conducting w ire carrying a steady current I is shaped as shown in thefigure below. All connected straight segments meet at right angles. Whatis the magnetic moment of the current loop?I abbaˆiˆjˆk(a)Iab (ˆj + ˆk)(b)Iab (ˆj − ˆk)(c)√2Iab (ˆj + ˆk)(d)Iab (ˆk − ˆj)11
The figure shows a wire with a loop in it. The current through the wire is 5.00 A and the diameter of the loop is 3.40 cm. What is the magnetic field at the center of the loop?
The rectangular loop of wire is being moved to the right at constant velocity. A constant current I flows in the long wire in the direction shown. What are the directions of the magnetic forces on the left-hand (L) and right-hand (R) sides of the loop?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.