Two voltmeters, Vright and Vleft, each with an internal resistance of 106 Ω are connected through wires of negligible resistance (see the circuit below). The “+" side of both voltmeters is up as shown. A changing magnetic field is present in the shaded area.
At a particular moment in time Vright reads -0.5 Volt (notice the − sign).
(a) What, at that moment, is the magnitude of the induced EMF (in Volts) in the circuit?
unanswered
(b) At that moment in time, what is the reading of Vleft? (make sure you have the correct sign!)
To find the magnitude of the induced EMF in the circuit, we need to use Faraday's Law of electromagnetic induction.
According to Faraday's Law, the induced EMF (ε) is directly proportional to the rate of change of magnetic flux (Φ) through the circuit. Mathematically, it can be expressed as:
ε = -dΦ/dt
Here, the negative sign indicates the direction of induced current. Since the voltmeter Vright reads -0.5 Volt, it means that the induced EMF is in the opposite direction to the conventional current flow.
Now, to find Vleft at that moment in time, we need to apply Kirchhoff's Voltage Law (KVL). According to KVL, the algebraic sum of the potential differences around any closed loop in a circuit must be equal to zero.
In this circuit, the loop consists of the voltmeters Vright and Vleft. Since both voltmeters have negligible resistance, the potential difference across each voltmeter will be equal to the induced EMF.
So, at that moment, the reading of Vleft will also be -0.5 Volt (since the potential difference across each voltmeter is the same).
Therefore, the reading of Vleft at that moment in time is -0.5 Volt.
To answer these questions, we need to understand the concept of electromagnetic induction. When a changing magnetic field is present in a closed loop of wire, it induces an electromotive force (EMF) in the loop, which causes a current to flow. In this circuit, the changing magnetic field is present in the shaded area.
(a) To determine the magnitude of the induced EMF in the circuit, we can use Faraday's Law of electromagnetic induction. Faraday's Law states that the magnitude of the induced EMF is equal to the rate of change of magnetic flux through the loop. Mathematically, it can be written as:
EMF = -dφ/dt
where EMF is the induced electromotive force and dφ/dt is the rate of change of magnetic flux.
In this case, we are given that Vright reads -0.5 Volt. Since the voltmeter has an internal resistance, we can assume that the voltmeter is measuring the potential difference across a portion of the wire with negligible resistance. Therefore, the potential difference measured by Vright is equal to the potential difference across the loop. This potential difference is -0.5 Volt.
The negative sign indicates that the direction of the induced EMF is opposite to the direction of the circuit current. This is due to the Lenz's Law, which states that the induced current will flow in such a way as to oppose the change causing it.
Therefore, the magnitude of the induced EMF in the circuit is 0.5 Volt.
(b) At that moment in time, the reading of Vleft can be determined by considering the fact that the voltmeter is connected across a portion of the wire. Since the potential difference across the entire loop is -0.5 Volt, we can assume that the potential difference across the wire between Vleft and the shaded area is the same. Therefore, the reading of Vleft will also be -0.5 Volt.