In a coil of resistance 100 ohms, a current is induced by changing the magnetic flux through it as s

Q: Why do we ignore the negative sign in Faraday's law here?

The negative sign in Faraday's law (Lenz's Law) indicates the direction of the induced emf. Since the question specifically asks for the 'magnitude' of the change in flux, we only need the absolute value, so we can safely drop the negative sign.

Q: What does the area under the current-time graph physically mean?

Current is the rate of flow of charge ($I = dq/dt$). When you integrate current over time (which is the area under the $I-t$ graph), you are calculating the total amount of electric charge ($Q$) that has flowed through the circuit during that time interval.

Q: Would the answer change if the graph was a curve instead of a straight line?

The fundamental relationship $\Delta\phi = R \times \text{Area}$ holds true regardless of the shape of the graph. As long as you can calculate the area under the curve (the total charge), you can find the change in flux. The straight line just makes calculating the area easier using basic geometry.

Q: Why can we pull the resistance $R$ out of the integral?

In this problem, the resistance of the coil is a constant $100\ \Omega$. In calculus, constant multipliers can always be factored out of an integral. If the resistance was changing with time (e.g., due to heating), we would have to keep it inside the integral.

Q: Is the total charge flowing through the coil dependent on how fast the flux changes?

No! This is a very important property. The equation $Q = \Delta\phi / R$ shows that the total charge depends only on the total change in flux and the resistance. It is completely independent of the time duration over which the flux changes.

Question

In a coil of resistance $100\ \Omega$, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is

Options:

(A) 225 Wb

(B) 250 Wb

(C) 275 Wb

(D) 200 Wb

Sigma Percentile · Accepted Answer

The relationship between a changing magnetic field and the electric current it induces is one of the most beautiful concepts in physics. In this problem, we are given a visual representation of this induced current over time, and our goal is to work backward to find the total change in magnetic flux that caused it.

I know that seeing a graph in an electromagnetic induction problem might seem intimidating at first, but let's take a breath. We have all the tools we need to decode it.

### Analyzing the Setup

Let's begin by carefully analyzing the information given to us. We have a coil with a constant resistance of $R = 100\ \Omega$.

Instead of an equation for the magnetic field, we are given a graph showing how the induced current $I$ in this coil changes with time $t$. Notice that the current starts at a maximum of $10\ 	ext{A}$ and drops linearly to $0\ 	ext{A}$ in exactly $0.5\ 	ext{s}$.

Our objective is to find the magnitude of the total change in magnetic flux, $\Delta\phi$, through the coil.

### The Master Equation

How do we connect the current flowing through the coil to the change in magnetic flux? We need to combine two fundamental laws of physics.

According to **Faraday's Law of Electromagnetic Induction**, the magnitude of the induced electromotive force (emf) is equal to the rate of change of magnetic flux:
$$ \mathcal{E} = \frac{d\phi}{dt} $$

Next, we apply **Ohm's Law**, which tells us that the induced current $I$ is simply this emf divided by the resistance $R$:
$$ I = \frac{\mathcal{E}}{R} = \frac{1}{R} \frac{d\phi}{dt} $$

Let's rearrange this equation to isolate the change in flux. By multiplying both sides by $R$ and $dt$, we get:
$$ d\phi = I \cdot R \cdot dt $$

To find the total change in flux $\Delta\phi$, we need to integrate this expression over the given time interval. Since the resistance $R$ is constant, we can pull it out of the integral:
$$ \Delta\phi = R \int I \, dt $$

### Geometrical Interpretation

Here is a crucial conceptual leap. What does the integral of current with respect to time, $\int I \, dt$, represent?

Physically, current is the rate of flow of charge ($I = \frac{dq}{dt}$). Therefore, integrating current over time gives us the **total charge $Q$** that has flowed through the circuit.

Graphically, the integral of a function is exactly the **area under its curve**. So, the area under our current-time graph represents this total charge $Q$.
$$ Q = \int I \, dt = 	ext{Area under } I-t 	ext{ graph} $$

### Final Calculation

Let's calculate this area. The shape under our graph is a right-angled triangle. The area of a triangle is half times its base times its height:
$$ 	ext{Area} = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} $$

Plugging in our values from the graph, the base is $0.5\ 	ext{s}$ and the height is $10\ 	ext{A}$:
$$ 	ext{Area} = \frac{1}{2} 	imes 0.5 	imes 10 = 2.5\ 	ext{C} $$

This means a total charge of $2.5\ 	ext{C}$ flowed through the coil. Finally, we substitute this area back into our flux equation:
$$ \Delta\phi = R 	imes 	ext{Area} $$
$$ \Delta\phi = 100 	imes 2.5 = 250\ 	ext{Wb} $$

The total change in magnetic flux is $250\ 	ext{Wb}$.

### The Way Forward

Before we wrap up, let's look at a powerful shortcut derived from our master equation. The total charge $Q$ flowing through a closed loop is always equal to the total change in magnetic flux divided by the resistance:
$$ Q = \frac{\Delta\phi}{R} $$

Notice that this formula does **not** depend on time! It doesn't matter if the flux changes in a millisecond or a year; as long as the total change in flux and the resistance are the same, the total charge that flows will be exactly the same. This is a favorite concept for JEE, so keep it in your toolkit!

Animated Solution for Physics - Electromagnetic Induction: In a coil of resistance $100 Ω$ , a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is

Select Answer:

Visualized Solution

Analyzing the Given Data

Faraday's and Ohm's Laws

Relating Flux to Current

Geometrical Interpretation

Calculating the Area

Total Change in Flux

Key Takeaway

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Analyzing the Setup

The Master Equation

Geometrical Interpretation

Final Calculation

The Way Forward

Similar Questions

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A coil having $n$ turns and resistance $R Ω$ is connected with a galvanometer of resistance $4 R Ω$ . This combination is moved for time $t$ seconds from a magnetic field $W_{1}$ weber to $W_{2}$ weber. The induced current in the circuit is

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At time $t = 0$ magnetic field of 1000 gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 gauss, in the next 5 s, then induced emf in the loop is

Select Answer:

Visualized Solution

Analyzing the Given Data

Faraday's and Ohm's Laws

Relating Flux to Current

Geometrical Interpretation

Calculating the Area

Total Change in Flux

Key Takeaway

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Analyzing the Setup

The Master Equation

Geometrical Interpretation

Final Calculation

The Way Forward

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The flux linked with a coil at any instant t is given by ϕ=10t2−50t+250. The induced emf at t=3 s is

The flux linked with a coil at any instant $t$ is given by $ϕ = 10 t^{2} - 50 t + 250$ . The induced emf at $t = 3 s$ is