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ElectricalElectromagnetics Field Theory

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What is curl of E for B = 15t?

$15$

$-15$

$7.5$

$-7.5$

Correct Answer

⚙️ TE • Technical Maxwell's EquationsElectricalElectromagnetics Field Theory

Option B

$-15$

Quick Summary:

Given: The magnetic flux density B is given as 15t, where t is time.

📐MAMath SolutionMaxwell's Equations

📋 Given

The magnetic flux density B is given as 15t, where t is time.

🔢 Formula Used

∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}∇×E=−∂t∂B​

🔢 Step-by-Step Solution

Identify Faraday's Law of Induction

Faraday's Law in differential form relates the curl of the electric field to the time rate of change of the magnetic flux density.

∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}∇×E=−∂t∂B​

Substitute the given magnetic field

We are given the magnetic flux density $\mathbf{B} = 15t$ . We substitute this into the formula.

∇×E=−∂(15t)∂t\nabla \times \mathbf{E} = -\frac{\partial (15t)}{\partial t}∇×E=−∂t∂(15t)​

Perform the differentiation

Differentiate $15t$ with respect to time $t$ . The derivative of $15t$ with respect to $t$ is $15$ .

∇×E=−(15)=−15\nabla \times \mathbf{E} = -(15) = -15∇×E=−(15)=−15

✅

B is correct because applying Faraday's Law, the curl of E is equal to the negative rate of change of B, which results in $-15$ .

Core Concepts Used

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Faraday's Law Maxwell's Equations Time-varying fields

💡 EXAM TIP

This concept is fundamental to the study of electromagnetic waves; always remember the negative sign, which represents Lenz's Law, ensuring energy conservation in inductive systems.

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